CIE A Level Maths: Probability & Statistics 2

Topic Questions

A LevelMaths: Probability & Statistics 2CIETopic Questions2. Statistical Distributions2.3 Continuous Random Variables

2.3 Continuous Random Variables

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1
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If f open parentheses x close parentheses is the probability density function for a random variable, X then the area under the graph of the probability density function must equal 1.

State, with a reason, whether each of the following could be graphs of probability density functions.

(i) eq1a (ii) eq1b
       
(iii) eq1c (iv) eq1d

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2
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If f open parentheses x close parentheses is the probability density function for a random variable, X which takes values in the interval open parentheses a comma b close parentheses then:

integral subscript a superscript b f left parenthesis x right parenthesis space d x equals 1

Use integration to decide whether the following functions could represent probability distribution functions.

(i)
f left parenthesis x right parenthesis equals open curly brackets table row cell 2 x space space space space space space space space space space space space space space space space space space space space space space space 0 less or equal than x less or equal than 1 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(ii)
f left parenthesis x right parenthesis equals open curly brackets table row cell 1 third x squared space space space space space space space space space space space space space space space space space space space space 1 less or equal than x less or equal than 4 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(iii)
f left parenthesis x right parenthesis equals open curly brackets table row cell 4 over 27 open parentheses x cubed plus 1 close parentheses space space space space space space space space minus 1 less or equal than x less or equal than 2 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

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3a
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The continuous random variable, X, has probability density function

f left parenthesis x right parenthesis equals open curly brackets table row cell 1 over 24 open parentheses x cubed plus 2 close parentheses space space space space space space space space space space space space space space space space 1 less or equal than x less or equal than 3 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

The value of P left parenthesis a less than X less than b right parenthesis is equal to

integral subscript a superscript b space f left parenthesis x right parenthesis space d x

Find P left parenthesis 1 less than X less than 2 right parenthesis.

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3b
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(i)
Explain why P left parenthesis X greater than 2.5 right parenthesis equals integral subscript 2.5 end subscript superscript 3 1 over 24 left parenthesis x cubed plus 2 right parenthesis d x

(ii)
Hence find P left parenthesis X greater than 2.5 right parenthesis

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3c
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State, with a reason, the value of

(i)
P left parenthesis 5 less than X less than 10 right parenthesis
(ii)
P left parenthesis X equals 2 right parenthesis

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4
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The graph below shows the probability density function of a continuous random variable, X.

e4

By finding relevant areas under the graph, find:

(i)
P left parenthesis 0 less than X less than 0.1 right parenthesis
(ii)
P left parenthesis X greater than 0.15 right parenthesis
(iii)
P left parenthesis 0.05 less than X less than 0.12 right parenthesis

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5a
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Evaluate the following definite integral, giving your answer in terms of k.

integral subscript 3 superscript 9 k left parenthesis x squared plus 3 right parenthesis space d x

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5b
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Hence find the value of k given that f open parentheses x close parentheses is a probability density function where

f left parenthesis x right parenthesis equals open curly brackets table row cell k open parentheses x squared plus 3 close parentheses space space space space space space space space space space space space space space space space space space 3 less or equal than x less or equal than 9 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

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6a
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The continuous random variable X has probability density function

f open parentheses x close parentheses equals space open curly brackets table row cell 3 over 16 open parentheses x squared minus 1 close parentheses space space space space space space space space space space space space space space space space space space space 2 less or equal than x less or equal than 3 space end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Find integral 3 over 16 space x left parenthesis x squared minus 1 right parenthesis space d x

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6b
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Hence find the value of E open parentheses X close parentheses using the formula 

E left parenthesis X right parenthesis equals integral x space f left parenthesis x right parenthesis space space d x

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6c
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Find integral 3 over 16 space x ² left parenthesis x squared minus 1 right parenthesis space d x

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6d
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Hence find the value of E open parentheses X squared close parentheses using the formula

E left parenthesis X squared right parenthesis equals integral x squared space f left parenthesis x right parenthesis space space d x

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6e
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Hence find the value of V a r left parenthesis X right parenthesis using the formula V a r left parenthesis X right parenthesis equals E left parenthesis X squared right parenthesis minus left parenthesis E left parenthesis X right parenthesis right parenthesis squared.

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7
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The mode of a continuous random variable X, where it exists, is a value for X where the probability density function is at its maximum.

State the mode, where it exists, of the continuous random variables which have probability density functions show in the graphs below.

(i) eq7a (ii) eq7b
       
(iii) eq7c (iv) eq7d

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8
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The median, m, of a continuous random variable X, is the value for X which splits the under the graph of the probability density function exactly in half, such that P left parenthesis X less than m right parenthesis equals 0.5.

Find the value of the median of the continuous random variables which have probability density functions show in the graphs below.

(i) eq8a (ii) eq8b
       
(iii) eq8c (iv) eq8d

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9a
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The continuous uniform (rectangular) distribution over the interval open parentheses a comma b close parentheses has probability density function

f left parenthesis x right parenthesis equals open curly brackets table row cell fraction numerator 1 over denominator b minus a end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space a less than x less than b end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

The random variable X follows a rectangular distribution over the interval open parentheses 2 comma 18 close parentheses.

The graph below shows the probability density function of X. Write down the values of a comma space b space and space c.

eq9

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9b
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Write down

(i)

the median of X

(ii)

the value of E open parentheses X close parentheses

(iii)
 P left parenthesis 14 less than X less than 18 right parenthesis

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1
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State, with a reason, whether the following functions could represent probability distribution functions.

(i)
f left parenthesis x right parenthesis equals open curly brackets table row cell 1 third left parenthesis x squared minus 2 right parenthesis space space space space space space space space space space 1 less or equal than x less or equal than 2 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(ii)
f left parenthesis x right parenthesis equals open curly brackets table row cell fraction numerator 4 over denominator x to the power of 5 space end fraction space space space space space space space space space space space space space space space space space space space space space space x greater or equal than 1 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(iii)
f left parenthesis x right parenthesis equals open curly brackets table row cell 2 over x squared space space space space space space space space space space space space space space space space space space space space space minus 1 less or equal than x less or equal than 2 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

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2a
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The continuous random variable, X, has probability density function

f left parenthesis x right parenthesis equals open curly brackets table row cell 1 fourth space x left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis space space space space space space space space space space space space space space space space 1 less or equal than x less or equal than k end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Find the exact value for k.

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2b
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Find P left parenthesis X less than E left parenthesis X right parenthesis right parenthesis.

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3a
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Harry is trying to draw specific lengths without measuring equipment. He draws a straight line and stops when he thinks it is 10 cm long. The actual length of Harry’s line, L cm, can be modelled by the probability density function

f left parenthesis l right parenthesis equals open curly brackets table row cell 20 over 421 open parentheses l minus 10.5 close parentheses to the power of 4 space space space space space space space space space space space space space space space space 8 less or equal than l less or equal than 12 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Estimate the lower quartile of the lengths of Harry’s lines.

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3b
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Harry tries to draw a 10 cm line 80 times.

Write down how many of Harry’s line you would expect to be less than the lower quartile.

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4a
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The diagram below shows the probability density function, f open parentheses x close parentheses, of a random variable X. f left parenthesis x right parenthesis equals k when 0 less or equal than x less or equal than a, otherwise f left parenthesis x right parenthesis equals 0.

hq4

Write down an expression for k in terms of a.

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4b
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Write down an expression for E open parentheses X close parentheses in terms of a.

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4c
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Find a simplified expression for V a r open parentheses X close parentheses in terms of a.

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4d
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Given that P left parenthesis X greater than 5 right parenthesis equals 0.6 find the value of a.

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5
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Find the mode of the continuous random variable, X, which has probability density function defined by:

(i)
f left parenthesis x right parenthesis equals open curly brackets table row cell 3 over 140 left parenthesis 5 minus 6 x minus x squared right parenthesis space space space space space space space space space space space space space space space space space space space space space space space minus 4 less or equal than x less or equal than 0 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(ii)
f left parenthesis x right parenthesis equals open curly brackets table row cell 3 over 34 left parenthesis 10 minus left parenthesis x minus 2 right parenthesis squared right parenthesis space space space space space space space space space space space space space space space space space space space space space space space 3 less or equal than x less or equal than 5 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(iii)
f left parenthesis x right parenthesis equals open curly brackets table row cell 1 half left parenthesis sin left parenthesis x right parenthesis plus cos left parenthesis x right parenthesis right parenthesis space space space space space space space space space space space space space space space space space space space space space space 0 less or equal than x less or equal than pi over 2 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

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6a
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The diagram below shows the probability density function, f open parentheses x close parentheses, of a random variable X.

hq6

Find the value of k.

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6b
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Find P left parenthesis X greater than 0.4 right parenthesis

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6c
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Find the median of X.

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7a
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The diagram below shows the probability density function, f open parentheses t close parentheses, of a random variable T.

hq7

For a less or equal than t less or equal than b comma space space f left parenthesis t right parenthesis equals 3 over 32 left parenthesis 10 t minus t squared minus 21 right parenthesis, elsewhere f left parenthesis t right parenthesis equals 0.

Find the values of a space and space b.

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7b
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State the value of E left parenthesis T right parenthesis and find V a r left parenthesis T right parenthesis.

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7c
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Find P left parenthesis negative 2 less or equal than T less or equal than 5 right parenthesis.

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7d
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Given that P left parenthesis 4 less or equal than T less or equal than 6 right parenthesis equals 11 over 16, find P left parenthesis T greater or equal than 6 right parenthesis.

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8a
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Geoff is taking part in a quiz where he has 4 seconds to answer each question. The time, in seconds, it takes Geoff to answer a question is denoted T which can be modelled by the probability density function, f open parentheses t close parentheses shown below.

hq8

Find the probability that Geoff answers a question in less than 2 seconds.

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8b
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Find the mean time it takes Geoff to answer a question.

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8c
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There are 10 questions in the quiz. Find the probability that for at least one of the questions Geoff takes longer than 3 seconds.

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1
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State, with a reason, whether each of the following could be graphs of probability density functions.

(i) mq1a (ii) mq1b
       
(iii) mq1c (iv)

mq1d

This is a semicircle.

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2
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State, with a reason, whether the following functions could represent probability distribution functions.

(i)
f left parenthesis x right parenthesis equals open curly brackets table row cell 3 left parenthesis x to the power of 5 minus 2 x cubed right parenthesis space space space space space space space space space space space space space space space space space space space space space space space 1 less or equal than x less or equal than 2 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(ii)
f left parenthesis x right parenthesis equals open curly brackets table row cell 3 over 8 open parentheses x plus 1 close parentheses squared space space space space space space space space space space space space space space space space space space space space space space space minus 1 less or equal than x less or equal than 1 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(iii)
f left parenthesis x right parenthesis equals open curly brackets table row cell 6 over x squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus 3 less or equal than x less or equal than negative 2 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

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3a
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The continuous random variable, X, has probability density function

f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell k open parentheses 3 x to the power of 4 minus 2 x close parentheses space space space space space space space space space space space space space space space space space space space space space space space 1 less or equal than x less or equal than 2 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Show that k equals 5 over 78.

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3b
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(i)
Find P left parenthesis 1.2 less than X less than 1.8 right parenthesis
(ii)
Find P left parenthesis X less than 1.5 right parenthesis

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4a
Sme Calculator2 marks

The continuous random variable X has probability density function

f left parenthesis x right parenthesis equals open curly brackets table row cell k open parentheses 2 plus square root of x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space 4 less or equal than x less or equal than 9 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Show that k equals 3 over 68.

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4b
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Find E open parentheses X close parentheses.

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4c
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Find V a r open parentheses X close parentheses.

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5
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Find the mode of the continuous random variable, X, which has probability density function defined by:

(i)
f left parenthesis x right parenthesis equals open curly brackets table row cell 2 over 21 x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 2 less or equal than x less or equal than 5 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(ii)
f left parenthesis x right parenthesis equals open curly brackets table row cell 1 over 9 left parenthesis 4 minus left parenthesis x plus 1 right parenthesis squared right parenthesis space space space space space space space space space space space space space space minus 2 less or equal than x less or equal than 1 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(iii)
f left parenthesis x right parenthesis equals open curly brackets table row cell 1 half sin open parentheses x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space 0 less or equal than x less or equal than pi end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

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6a
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The continuous random variable, X, has probability density function

f left parenthesis x right parenthesis equals open curly brackets table row cell 5 over 62 square root of x cubed end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space 1 less or equal than x less or equal than 4 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Find an expression, in terms of m, for

integral subscript 1 superscript m 5 over 62 square root of x cubed space end root space d x

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6b
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Hence find the median of X.

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7a
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The continuous random variable X follows a continuous uniform distribution over the interval left parenthesis 3 comma 28 right parenthesis so that its probability density function is

f left parenthesis x right parenthesis equals open curly brackets table row cell 1 over 25 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 3 less or equal than x less or equal than 28 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Find

(i)
P left parenthesis X less than 15 right parenthesis
(ii)
P left parenthesis 20 less than X less than 30 right parenthesis
(iii)
P left parenthesis X equals 25 right parenthesis

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7b
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(i)
Write down E open parentheses X close parentheses.
(ii)
Find V a r space open parentheses X close parentheses.

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8a
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May calls her nephew Peter each day to check on him. May enjoys talking so much so that Peter has programmed his phone to disconnect from a call after 15 minutes. The length of a call, in minutes, between May and Peter is denoted by the continuous random variable T. It is modelled by the probability density function

f left parenthesis t right parenthesis equals open curly brackets table row cell 1 over 1125 t squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 0 less than t less than 15 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Find the mean length of a call.

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8b
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Find the standard deviation of the lengths of calls.

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8c
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Find the probability that a random call between May and Peter will last longer than 10 minutes.

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8d
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Out of the next 100 calls between May and Peter, estimate the number of them that will last less than 10 minutes.

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9a
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The mass, in kilograms, of a wrestler in a local club is denoted by the continuous random variable M. It is modelled by the probability density function

f left parenthesis m right parenthesis equals open curly brackets table row cell k minus 1 over 9000 left parenthesis m minus 80 right parenthesis ² space space space space space space space space space space space space space space space space space space space space space space 70 less than m less than 100 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Show that k equals 4 over 90.

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9b
Sme Calculator2 marks

Wrestlers who weigh less than 85 kg can enter Heavy Middleweight competitions.

Find the probability that a randomly selected wrestler can enter Heavy Middleweight competition.

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9c
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Wrestlers who weigh less than 80 kg can enter Middleweight competitions.

Given that a wrestler can enter Heavy Middleweight competitions, find the probability that they can also enter Middleweight competitions.

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1a
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The diagram below shows the probability density function, f open parentheses x close parentheses, of a random variable X.

vhq--

Find the value of k.

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1b
Sme Calculator2 marks

Find P left parenthesis X less or equal than 6 right parenthesis.

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1c
Sme Calculator4 marks

Find the interquartile range of X.

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2a
Sme Calculator2 marks

A computer takes T seconds to start up. The random variable, T, can be modelled by the probability distribution, f open parentheses t close parentheses, shown below.

vhq

For a less or equal than t less or equal than b comma space space f left parenthesis t right parenthesis equals fraction numerator 1 over denominator 8 pi end fraction square root of left parenthesis 12 t minus t squared minus 20 right parenthesis end root, elsewhere f left parenthesis t right parenthesis equals 0.

Find the values of a space and space b.

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2b
Sme Calculator1 mark

State the mean time for the computer to start up.

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2c
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Find probability that the computer takes between 6 and 15 seconds to start up.

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2d
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Given that P left parenthesis 4 less or equal than T less or equal than 8 right parenthesis equals square root of fraction numerator 3 over denominator 2 pi end fraction end root plus 1 third, find the exact value of P left parenthesis T greater than 8 right parenthesis.

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3a
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The diagram below shows the probability density function, g open parentheses y close parentheses, of a random variable Y.

vhq3

Find the value of k.

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3b
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Find P left parenthesis 5 less than Y less than 6 right parenthesis.

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3c
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Find the median of Y.

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3d
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Find E left parenthesis Y right parenthesis.

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4a
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The random variable, U, follows a continuous uniform distribution on the interval left square bracket a comma b right square bracket. The probability density function g is defined by:

g left parenthesis u right parenthesis equals open curly brackets table row cell k space space space space space space space space space space space space space space space space space space space space space a less or equal than u less or equal than b end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

(i)
Write down an expression for k in terms of a space and space b.
(ii)
Write down an expression for E left parenthesis U right parenthesis in terms of a space and space b.
(iii)
Show that V a r left parenthesis U right parenthesis equals fraction numerator left parenthesis b minus a right parenthesis ² over denominator 12 end fraction.

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4b
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Given that E left parenthesis U right parenthesis equals 8.5 space and space V a r left parenthesis U right parenthesis equals 6.75, find the values of a space and space b.

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5a
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A meeting at a company lasts T hours. The random variable T can be modelled by the probability density function

f left parenthesis t right parenthesis equals open curly brackets table row cell 3 over 4 t open parentheses t minus 2 close parentheses squared space space space space space space space space space space space space space space space space space 0 less or equal than t less or equal than 2 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Find the mean time of a meeting and find the standard deviation of times.

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5b
Sme Calculator4 marks

Show that the median length of a meeting is 0.771 hours, correct to 3 significant figures.

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5c
Sme Calculator4 marks

By using differentiation, find the mode of the times.

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6a
Sme Calculator3 marks

The continuous random variable X has probability density function

f left parenthesis x right parenthesis equals open curly brackets table row cell k over x cubed space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 2 less or equal than x less or equal than a end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Show that k equals fraction numerator 8 a ² over denominator a squared minus 4 end fraction.

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6b
Sme Calculator5 marks

Given that E left parenthesis X right parenthesis equals 20 over 7 show that a equals 5 and hence find the value of k.

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6c
Sme Calculator3 marks

Find the exact value of V a r left parenthesis X right parenthesis.

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7a
Sme Calculator1 mark

Paul is travelling around France to try to find the perfect baguette. The random variable C represents the cost, in euros, of a baguette. It can be modelled by the probability density function

f left parenthesis c right parenthesis equals open curly brackets table row cell 1 over 36 open parentheses 14 c minus c squared minus 40 close parentheses space space space space space space space space space space space space space space space space 4 less or equal than c less or equal than 10 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Sketch the graph of y equals f left parenthesis c right parenthesis.

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7b
Sme Calculator1 mark

Explain why the mean price of a baguette is €7.

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7c
Sme Calculator2 marks

Given that P left parenthesis C less than 9 right parenthesis equals 25 over 27, find the value of P left parenthesis 5 less than C less than 9 right parenthesis.

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