Cumulative Distribution Function With Example
In this class, We discuss the Cumulative Distribution Function With an Example.
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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(X<= x) = Σxi <= 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 < 0
= 1/8 if 0 <= x <1
= 4/8 if 1 <= x < 2
= 7/8 if 2 <= x < 3
= 1 if 3 <= x < infinite
Similarly, we can convert the cumulative distribution function to the probability mass function.
Example:
The cumulative distribution function F(x)
= 0 if x < -2
= 0.2 if -2 <= x < 0
= 0.7 if 0 <= x < 2
= 1 if 2<= 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).
