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.

Cumulative Distribution Function With Example1

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).

Cumulative Distribution Function With Example2