DP IB Maths: AA SL

Revision Notes

4.4.1 Discrete Probability Distributions

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Discrete Probability Distributions

What is a discrete random variable? 

  • A random variable is a variable whose value depends on the outcome of a random event
    • The value of the random variable is not known until the event is carried out (this is what is meant by 'random' in this case)
  • Random variables are denoted using upper case letters (X comma space Y, etc )
  • Particular outcomes of the event are denoted using lower case letters (x comma space y, etc)
  • straight P left parenthesis X equals x right parenthesis means "the probability of the random variable X taking the value x"
  • A discrete random variable (often abbreviated to DRV) can only take certain values within a set
    • Discrete random variables usually count something
    • Discrete random variables usually can only take a finite number of values but it is possible that it can take an infinite number of values (see the examples below)
  • Examples of discrete random variables include:
    • The number of times a coin lands on heads when flipped 20 times
      • this has a finite number of outcomes: {0,1,2,…,20}
    • The number of emails a manager receives within an hour
      • this has an infinite number of outcomes: {1,2,3,…}
    • The number of times a dice is rolled until it lands on a 6
      • this has an infinite number of outcomes: {1,2,3,…}
    • The number that a dice lands on when rolled once
      • this has a finite number of outcomes: {1,2,3,4,5,6}

What is a probability distribution of a discrete random variable?

  • A discrete probability distribution fully describes all the values that a discrete random variable can take along with their associated probabilities
    • This can be given in a table
    • Or it can be given as a function (called a discrete probability distribution function or "pdf")
    • They can be represented by vertical line graphs (the possible values for along the horizontal axis and the probability on the vertical axis)
  • The sum of the probabilities of all the values of a discrete random variable is 1
    • This is usually written sum straight P left parenthesis X equals x right parenthesis equals 1
  • A discrete uniform distribution is one where the random variable takes a finite number of values each with an equal probability
    • If there are n values then the probability of each one is 1 over n

4-1-1-discrete-probability-distributions-diagram-1

How do I calculate probabilities using a discrete probability distribution? 

  • First draw a table to represent the probability distribution
    • If it is given as a function then find each probability
    • If any probabilities are unknown then use algebra to represent them
  • Form an equation using sum straight P left parenthesis X equals x right parenthesis equals 1
    • Add together all the probabilities and make the sum equal to 1
  • To find straight P left parenthesis X equals k right parenthesis
    • If k is a possible value of the random variable X then straight P left parenthesis X equals k right parenthesis will be given in the table
    • If k is not a possible value then straight P left parenthesis X equals k right parenthesis equals 0
  • To find straight P left parenthesis X less or equal than k right parenthesis
    • Identify all possible values, x subscript i, that X can take which satisfy x subscript i less or equal than k
    • Add together all their corresponding probabilities
    • straight P left parenthesis X less or equal than k right parenthesis equals sum for x subscript i less or equal than k of straight P left parenthesis X equals x subscript i right parenthesis
    • Some mathematicians use the notation straight F left parenthesis x right parenthesis to represent the cumulative distribution
      • straight F left parenthesis x right parenthesis equals straight P left parenthesis X less or equal than x right parenthesis
  • Using a similar method you can find straight P left parenthesis X less than k right parenthesisstraight P left parenthesis X greater than k right parenthesis and straight P left parenthesis X greater or equal than k right parenthesis
  • As all the probabilities add up to 1 you can form the following equivalent equations:
    • straight P left parenthesis X less than k right parenthesis plus straight P left parenthesis X equals k right parenthesis plus straight P left parenthesis X greater than k right parenthesis equals 1
    • straight P left parenthesis X greater than k right parenthesis equals 1 minus straight P left parenthesis X less or equal than k right parenthesis
    • straight P left parenthesis X greater or equal than k right parenthesis equals 1 minus straight P left parenthesis X less than k right parenthesis

How do I know which inequality to use? 

  • straight P left parenthesis X less or equal than k right parenthesis would be used for phrases such as:
    • At most , no greater than , etc
  • straight P left parenthesis X less than k right parenthesis would be used for phrases such as:
    • Fewer than
  • straight P left parenthesis X greater or equal than k right parenthesis would be used for phrases such as:
    • At least , no fewer than , etc
  • straight P left parenthesis X greater than k right parenthesis would be used for phrases such as:
    • Greater than , etc

Worked example

The probability distribution of the discrete random variable X is given by the function

straight P left parenthesis X equals x right parenthesis equals stretchy left curly bracket table row cell k x ² end cell row 0 end table blank table attributes columnalign left end attributes row cell x equals negative 3 comma blank minus 1 comma blank 2 comma blank 4 end cell row cell otherwise. end cell end table

a)
Show that k equals 1 over 30.

4-4-1-ib-ai-aa-sl-discrete-pd-a-we-solution

b)
Calculate straight P left parenthesis X less or equal than 3 right parenthesis.

4-4-1-ib-ai-aa-sl-discrete-pd-b-we-solution

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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.