Solve Cumulative Probability Distribution Function

Computation of probability of events is called as probability distribution.  It can give all possible outputs in a single process.  Cumulative probability distribution function of a random variable X is defined as F(x) = P(X <= x) = sum_(x)I <= x)p(x_i): (-oo < x oo).  Let us discuss about the topic of “solve cumulative probability distribution function” in the following below with some related problems.

Example Problems – Solve Cumulative Probability Distribution Function

Example problem 1

A random variable Y has the following probability mass function.  Find the value of b and evaluate `P(Y lt= 2)` .

Y	0	1	2	3	4
P(Y = y)	10b	20b	30b	40b	50b

Solution:

P(Y = y) is the probability mass function `sum_(y = 0)^4 (Y = y) = 1`

P(Y=0) + P(Y = 1) + P(Y = 2) + P(Y = 3) + P(Y = 4) = 1

10b + 20b + 30b + 40b + 50b = 1

150b = 1

b = `1/150`

`P(Y lt= 2) = P(Y = 0) + P(Y = 1)`

                = `10/150 + 20/150 + 30/150`

                = `(10 + 20 + 30)/150`

                = `60/150`

                = `2/5`

Answer:

The value of b is `1/150`

`P(Y lt= 2) = 2/5`

Example problem 2

Find out probability mass function and cumulative distribution function for getting ‘4’s when two dice are thrown.

Solution:

Let us consider Y be the random variable of getting number of ‘4’s.

First, we are going to find the probability mass function

Therefore, Y can take the values 0, 1, 2.

P(no ‘4’) = P(Y = 0) = `25/36`

P(one ‘4’) = P(Y = 1) = `10/36`

P(two ‘4’) = P(Y = 2) = `1/36`

Probability mass function

y	0	1	2
P(Y = y)	25/36	10/36	1/36

Now, we are going to find the cumulative probability distribution function

Formula:  `F(x) = sum_(x_i = -oo)^x P(X = x_i)`

Y = 0, F(0) = P(Y = 0) = 25/36

Y = 1, `F(1) = sum_(i = -oo)^1 P(Y = y_i)`

                = P(Y = 0) + P(Y = 1)

                = `25/36 + 10/36`

                = `35/36`

Y = 2, `F(2) = sum_(i = -oo)^2 P(Y = y_i)`

                = P(Y = 0) + P(Y = 1) + P(Y = 2)

                = `25/36 + 10/36 + 1/36`

                = `36/36`

                = 1

Cumulative distribution function

y	0	1	2
F(Y)	25/36	35/36	1

Answer:

Probability mass function

y	0	1	2
P(Y = y)	25/36	10/36	1/36

Cumulative distribution function

y	0	1	2
F(Y)	25/36	35/36	1

Practicing Problems – Solve Cumulative Probability Distribution Function

Practicing problem 1

Find the cumulative distribution function for the given probability mass function

y	0	1	2	3
P(Y = y)	2a	4a	6a	8a

Find the value of a and `P(Y lt= 3)`

Answer:

The value of a is `1/20`

`P(Y lt= 3) = 1`

Practicing problem 2

Find the cumulative distribution function for getting number of heads when three coins are tossed once.

Answer:

Cumulative distribution function

y	0	1	2	3
F(y)	1/8	1/2	7/8	1