CIE A Level Maths: Probability & Statistics 2

Revision Notes

A LevelMaths: Probability & Statistics 2CIERevision Notes2. Statistical Distributions2.2 Linear Combinations of Random Variables2.2.1 Linear Combinations of Random Variables

2.2.1 Linear Combinations of Random Variables

Test Yourself

aX+b

How are the mean and variance of X related to the mean and variance of aX + b?

  • If a and b are constants then the following results are true
    • E(aX + b) = aE(X) + b
    • Var(aX + b) = a² Var(X)
  • Note that the mean is affected by multiplication and addition whereas addition does not change the variance
  • The factor of a²  includes the squared because the values of X are squared in the calculation
    • You could try and use the first result and the formula for variance to verify the second result
  • Remember a subtraction can be written as an addition
    • X – b can be written as X + (-b)
  • And division can be written as a multiplication
    • begin mathsize 16px style x over a end stylecan be written as begin mathsize 16px style 1 over a X end style

What does the distribution of aX + b look like?

  • A linear function is applied to each value of X
  • The graphical representation of aX + b is a linear transformation (a translation and a stretch) of the graphical representation of X
  • If X follows a normal distribution then aX + b will also follow a normal distribution
    • If X ~ N(μ, σ²) then aX + b ~ N(aμ + b, a²σ²)
  • If X follows a binomial, geometric or Poisson distribution then aX + b will no longer follow the same type of distribution

Worked example

X is a random variable such that E left parenthesis X right parenthesis equals 5 and Var left parenthesis X right parenthesis equals 4.

Find the value of:

(i)
straight E left parenthesis 3 X plus 5 right parenthesis
(ii)
Var left parenthesis 3 X plus 5 right parenthesis
(iii)
Var left parenthesis 2 minus X right parenthesis

2-2-1-ax--b-we-solution-1-2

aX + bY

How are the means and variances of X and Y related to the mean and variance of X + Y ?

  • If X and Y are two random variables then X + Y is the random variable whose values are the sums of each pair containing one value of X and one value of Y
  • E(X + Y) = E(X) + E(Y)
    • this is true for any random variables X and Y
    • Note that E(X – Y) = E(X) - E(Y) (see below for more information)
  • Var(X + Y) = Var(X) + Var(Y)
    • this is true if X and Y are independent
    • Note that Var(X - Y) = Var(X) + Var(Y) (see below for more information)

What does the distribution of X + Y  look like?

  • If X and Y are two independent Poisson distributions then X + Y is also a Poisson distribution
    • If begin mathsize 16px style X to the power of tilde P o left parenthesis lambda right parenthesis end style and size 16px Y to the power of size 16px tilde size 16px P size 16px o size 16px left parenthesis size 16px mu size 16px right parenthesis then begin mathsize 16px style X plus Y to the power of tilde P o left parenthesis lambda plus mu right parenthesis end style
  • If X and Y are two independent normal distributions then X + Y is also a normal distribution
    • If begin mathsize 16px style X to the power of tilde N open parentheses mu subscript 1 comma sigma subscript 1 squared close parentheses end style and Y to the power of size 16px tilde size 16px N begin mathsize 16px style stretchy left parenthesis mu subscript 2 comma sigma subscript 2 squared stretchy right parenthesis end style then X plus Y to the power of tilde N left parenthesis mu subscript 1 plus mu subscript 2 size 16px comma size 16px sigma subscript size 16px 1 to the power of size 16px 2 size 16px plus size 16px sigma subscript size 16px 2 to the power of size 16px 2 size 16px right parenthesis

What does the distribution of aX + bY look like?

  • If X and Y are random variables and a and b are two constants we can combine the results for aX + b and X + Y
  • E(aX + bY) = aE(X) + bE(Y)
    • this is true for any random variables X and Y
  • Var(aX + bY) = a²Var(X) + b²Var(Y)
    • this is true if X and Y are independent
  • Note that b is squared for the variance so we have
    • E(aX - bY) = aE(X) - bE(Y)
    • Var(aX - bY) = a²Var(X) + b²Var(Y)
      • Notice that the variances of aX + bY and aX – bY are the same
  • If X and Y are two independent normal distributions then aX + bY is also a normal distribution
    • If X ~ N(μ1, σ1²) and Y ~ N(μ2, σ2²) then aX ± bY ~ N(aμ1 ± bμ2, a²σ1² + b²σ2²)
  • Note that aX + bY is no longer Poisson even if X and Y are Poisson
    • This holds provided a and b are not 0 or 1

Worked example

 X and Y are independent random variable such that

straight E left parenthesis X right parenthesis space equals space 5 space and space Var space left parenthesis X right parenthesis space equals space 3

 straight E left parenthesis Y right parenthesis equals negative 2 space and space Var left parenthesis Y right parenthesis equals 4.

 Find the value of:

(i)
straight E left parenthesis 2 X plus 5 Y right parenthesis
(ii)
Var left parenthesis 2 X plus 5 Y right parenthesis
(iii)
Var left parenthesis 4 X minus Y right parenthesis

2-2-1-ax--by-we-solution-1

Linear Combinations

For a given random variable X, what is the difference between 2X and X1 + X2?

  • 2X means one observation of X is taken and then doubled
  • X1 + X2 means two observations of X are taken and added together
  • 2X and X1 + X2 both have the same expected value of 2E(X)
  • 2X and X1 + X2 have different variances
    • Var(2X) = 2²Var(X) = 4Var(X)
    • Var(X1 + X2) = 2Var(X)
  • Imagine X could take the values 0 and 1
    • 2X could then take the values 0 and 2 (2 × 0 = 0 and 2 × 1 = 2)
    • X1 + X2 could then take the values 0, 1 and 2 (0 + 0 = 0, 0 +1 = 1, 1 + 1 = 2)
  • Sometimes questions may describe the variables in context
    • The mass of a carton of half a dozen eggs is the mass of the carton plus the mass of the 6 individual eggs and can be modelled using the random variable
    • C + E1 + E2 + E3 + E4 + E5 + E6 where
      • C is the mass of a carton
      • E is the mass of an egg
    • It is not C + 6E because the masses of the 6 eggs could be different

How do I use linear combinations of normal random variables to find probabilities?

  • If the random variables are normally distributed and independent you might be asked to find probabilities such as
    • P(X1 + X2 + X3 > 2Y + 5)
    • This could be given in words
      • Find the probability that the mass of three chickens (X) is more than 5 kg heavier than double the mass of a turkey (Y)
  • To solve these problems:
    • STEP 1: Rearrange the inequality to get all the random variables on one side
      • P(X1 + X2 + X3 – 2Y > 5)
    • STEP 2: Find the mean and variance of the combined normal random variable
      • μ = E(X1 + X2 + X3 – 2Y) = E(X1) + E(X2) + E(X3) - 2E(Y)
      • σ² = Var(X1 + X2 + X3 – 2Y) = Var(X1) + Var(X2) + Var(X3) + 2² Var(X1)
    • STEP 3: Find the required probability using the combined normal distribution
      • X1 + X2 + X3 – 2Y ~ N(μ, σ²)
      • Use z-values and the table of values

Worked example

X tilde N left parenthesis 10 comma space 4 squared right parenthesis space and space Y tilde N left parenthesis negative 5 comma space 8 squared right parenthesis

Find straight P left parenthesis 3 X greater than 2 Y plus 50 right parenthesis

2-2-1-linear-combs-we-solution-1

Exam Tip

  • Be careful with negatives!

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    Dan

    Author: Dan

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.