Wize AP Statistics Textbook > Random Variables and Probability Distributions

Cumulative Distribution Function

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Cumulative Distribution Function (CDF)
A cumulative distribution function F(x)F(x)F(x) assigns a "running total" probability for the each value of XXX.

For discrete random variables, we use a cumulative distribution table.
    x       F(x)=P(X≤x)  x1P(x1)x2P(x1)+P(x2)⋮⋮1\begin{array}{c|c}
\ \ \ \ x\ \ \ \ \ &\ \ F(x)=P(X\le x)\ \ \\
\hline
x_1&P(x_1)\\
x_2&P(x_1)+P(x_2)\\
\vdots&\vdots\\
&1


\end{array}    x     x1​x2​⋮​  F(x)=P(X≤x)  P(x1​)P(x1​)+P(x2​)⋮1​​

Wize Tip
The column xxx must contain all possible values of XXX
The first F(x)F\left(x\right)F(x) value in the CDF is the same as in the PDF
The F(xn)F\left(x_n\right)F(xn​) values must increase as we go down the column
The last F(x)F\left(x\right)F(x) value in the CDF must equal 1
Given a PDF you can always get a CDF (vice versa)
Don't forget about calculating probabilities using the complement rule!

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Example: Cumulative Distribution Function (CDF)
Let XXX be a discrete random variable with values all possible even integers. The following cumulative distribution information is given:


a) Find P(X>4)P\left(X>4\right)P(X>4)

P(X>4)=P(X=6,8,10,12,14,… )P\left(X>4\right)=P\left(X=6,8,10,12,14,\dots\right)P(X>4)=P(X=6,8,10,12,14,…)

Since we don’t have information for X>10X > 10X>10, we will use the complement

P(X>4)=1−P(X≤4)=1−0.25=0.75P\left(X>4\right)=1−P\left(X\le4\right)=1−0.25=0.75P(X>4)=1−P(X≤4)=1−0.25=0.75

b) Find P(2≤X≤8)P\left(2\le X\le8\right)P(2≤X≤8)

P(2≤X≤8)=P(X=2,4,6,8)=P(𝑋≤8)−P(𝑋≤0)=0.59−0.05=0.54P\left(2\le X\le8\right)=P\left(X=2,4,6,8\right)=P\left(𝑋\le8\right)−P\left(𝑋\le0\right)=0.59−0.05=0.54P(2≤X≤8)=P(X=2,4,6,8)=P(X≤8)−P(X≤0)=0.59−0.05=0.54

c) Find P(X=6)P\left(X=6\right)P(X=6)

The answer is not 0.39!

P(X=6)=F(6)−F(4)=0.39−0.25=0.14P\left(X=6\right)=F\left(6\right)-F\left(4\right)=0.39-0.25=0.14P(X=6)=F(6)−F(4)=0.39−0.25=0.14

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Or, you could have created the probability distribution function first, and then answered the questions.

The cumulative distribution function (cdf) of a discrete random variable that only takes on integer values is given by F(x)F(x)F(x). 

Suppose F(−1)=0.61F(-1)=0.61F(−1)=0.61, F(2)=0.75F(2)=0.75F(2)=0.75, F(3)=0.80F(3)=0.80F(3)=0.80,F(4)=0.87F(4)=0.87F(4)=0.87 and F(5)=0.95F(5)=0.95F(5)=0.95

Find P(2<X≤3)+P(X≥4)P\left(2<X\le3\right)+P\left(X\ge4\right)P(2<X≤3)+P(X≥4)