Cumulative Distribution Function With Example
In this class, We discuss the Cumulative Distribution Function With an Example.
The reader should have prior knowledge of the probability distribution function. Click Here.
We take an example and understand the concept of the cumulative distribution function.
Example:
Toss three coins.
Random variable X = number of heads.
The below table shows the probability distribution for the random variable X.
f(x) is a function that provide the probability values for the random variable X = x.
Cumulative distribution function:
The name itself says cumulative. I.e. add the previous probability values.
The cumulative distribution function is shown as F(x).
F(x) = P(Xlt= x) = Σxi lt= x (fxi)
Important: You can write the cumulative distribution function if you know the probability mass function.
Example:
The below table shows the probability distribution for three coin tosses.
Find the cumulative distribution function.
F(x) = 0 if x lt 0
= 1/8 if 0 lt= x lt1
= 4/8 if 1 lt= x lt 2
= 7/8 if 2 lt= x lt 3
= 1 if 3 lt= x lt infinite
Similarly, we can convert the cumulative distribution function to the probability mass function.
Example:
The cumulative distribution function F(x)
= 0 if x lt 2
= 0.2 if 2 lt= x lt 0
= 0.7 if 0 lt= x lt 2
= 1 if 2lt= x
Solution:
f(x1) = f(2) = 0.2
Our first random variable value X = 2
Second random variable value X = 0
f(x1) + f(x2) = f(2) + f(0) = 0.7
f(0) = 0.7 – 0.2 = 0.5
Third random variable value X = 2
f(x1) + f(x2) + f(x3) = 1
f(2) + f(0) + f( 2) = 1
f(2) = 1 – 0.7
f(2) = 0.3
The below table shows the probability distribution for the cumulative distribution function F(x).

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