AP®StatisticsCollege BoardRevision NotesUnit 4: Probability, Random Variables & Probability DistributionsDiscrete Random VariablesCumulative Probability Distributions for Discrete Random Variables

Cumulative Probability Distributions for Discrete Random Variables (College Board AP® Statistics)

Revision Note

Dan Finlay

Expertise

Maths Lead

Discrete cumulative probability distributions

What is a discrete cumulative probability distribution?

  • A discrete cumulative probability distribution shows the probability that a discrete random variable is less than or equal to each of its possible values

  • A discrete cumulative probability distribution can be given as either a table or a function

  • To find the cumulative probability P open parentheses X less or equal than x close parentheses

    • identify the values of the random variable that are less than or equal to the x

    • add together the probabilities of these values

  • The cumulative probability of the smallest value is always equal to the probability of that value

    • e.g. if X can only take the values 0, 2 or 4, then P open parentheses X less or equal than 0 close parentheses equals P open parentheses X equals 0 close parentheses

  • The cumulative probability of the largest value is always equal to 1

    • e.g. if X can only take the values 0, 2 or 4, then P open parentheses X less or equal than 4 close parentheses equals 1

Diagram showing probability and cumulative probability distributions. First probability is 0.5, second is 0.3, third is 0.2. Cumulative sums are 0.5, 0.8, and 1.
Example of a cumulative probability distribution

How can I find probabilities using a cumulative probability distribution?

  • To find P open parentheses X less or equal than x close parentheses

    • if X can take the value x

      • read P open parentheses X less or equal than x close parentheses directly from the distribution

    • if X cannot take the value x

      • find the biggest value of X that is less than x

      • find the cumulative probability of this value

        • e.g. if X can take the values, 1, 2, 3 or 5 then P open parentheses X less or equal than 4 close parentheses equals P open parentheses X less or equal than 3 close parentheses

  • To find P open parentheses X less than x close parentheses

    • find the biggest value of X that is less than x

    • find the cumulative probability of this value

      • e.g. if X can take the values, 1, 2, 3 or 5 then P open parentheses X less than 5 close parentheses equals P open parentheses X less or equal than 3 close parentheses

  • To find P open parentheses X greater than x close parentheses use the identity

    • P open parentheses X greater than x close parentheses equals 1 minus P open parentheses X less or equal than x close parentheses

  • To find P open parentheses X greater or equal than x close parentheses use the identity

    • P open parentheses X greater or equal than x close parentheses equals 1 minus P open parentheses X less than x close parentheses

  • To find P open parentheses a less or equal than X less or equal than b close parentheses use the identity

    • P open parentheses X less or equal than b close parentheses minus P open parentheses X less than a close parentheses

      • Note that a is not included in the second inequality

  • To find P open parentheses X equals x close parentheses

    • find the biggest value of X that is less than x

    • find the cumulative probability of this value

    • subtract this probability from P open parentheses X less or equal than x close parentheses

      • e.g. if X can take the values, 1, 2, 3 or 5 then P open parentheses X equals 3 close parentheses equals P open parentheses X less or equal than 3 close parentheses minus P open parentheses X less or equal than 2 close parentheses

Worked Example

X is a discrete random variable that can take any positive integer value. The cumulative probability distribution is given by the function P open parentheses X less or equal than x close parentheses equals fraction numerator 1 over denominator x plus 1 end fraction, where x is a positive integer.

(a) Find P open parentheses X equals 1 close parentheses.

Answer:

1 is the first positive integer and so is the smallest value that X can take

Therefore P open parentheses X less or equal than 1 close parentheses equals P open parentheses X equals 1 close parentheses

Substitute x equals 1 into the function

table row cell P open parentheses X less or equal than 1 close parentheses end cell equals cell fraction numerator 1 over denominator 1 plus 1 end fraction end cell row blank equals cell 1 half end cell end table

P open parentheses X equals 1 close parentheses equals 1 half

(b) Find P open parentheses X greater or equal than 5 close parentheses.

Answer:

Use the identity P open parentheses X greater or equal than x close parentheses equals 1 minus P open parentheses X less than x close parentheses

P open parentheses X greater or equal than 5 close parentheses equals 1 minus P open parentheses X less than 4 close parentheses

3 is the largest value that X can take that is also less than 4

P open parentheses X less than 4 close parentheses equals P open parentheses X less or equal than 3 close parentheses

Substitute x equals 3 into the function

table row cell P open parentheses X less or equal than 3 close parentheses end cell equals cell fraction numerator 1 over denominator 3 plus 1 end fraction end cell row blank equals cell 1 fourth end cell end table

This is also P open parentheses X less than 4 close parentheses so substitute into P open parentheses X greater or equal than 5 close parentheses equals 1 minus P open parentheses X less than 4 close parentheses

table row cell P open parentheses X greater or equal than 5 close parentheses end cell equals cell 1 minus 1 fourth end cell row blank equals cell 3 over 4 end cell end table

P open parentheses X greater or equal than 5 close parentheses equals 3 over 4

(c) What is the value of P open parentheses X less or equal than 3.5 close parentheses?

(A) 0

(B) 1 fourth

(C) 2 over 9

(D) 1 fifth

Answer:

X cannot take the value of 3.5

3 is the largest value that X can take that is also less than 3.5

Substitute x equals 3 into the function

table row cell P open parentheses X less or equal than 3 close parentheses end cell equals cell fraction numerator 1 over denominator 3 plus 1 end fraction end cell row blank equals cell 1 fourth end cell end table

The correct answer is B

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    Dan Finlay

    Author: Dan Finlay

    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.