Mathematical Expectation: Solved Example Problems

Example 6.12

Determine the mean and variance of the random variable X having the following probability distribution.

Solution:

Therefore, the mean and variance of the given discrete distribution are 6.56 and 7.35 respectively.

Example 6.13

Six men and five women apply for an executive position in a small company. Two of the applicants are selected for an interview. Let X denote the number of women in the interview pool. We have found the probability mass function of X.

How many women do you expect in the interview pool?

Solution:

Expected number of women in the interview pool is

Example 6.14

Determine the mean and variance of a discrete random variable, given its distribution as follows:

Solution

From the given data, you first calculate the probability distribution of the random variable. Then using it you calculate mean and variance.

Example 6.15

The following information is the probability distribution of successes.

Determine the expected number of success.

Solution

Expected number of success is

Therefore, the expected number of success is 0.5. Approximately one success.

Example 6.16

An urn contains four balls of red, black, green and blue colours. There is an equal  probability of getting any coloured ball. What is the expected value of getting a blue ball  out of 30 experiments with replacement?

Solution

Probability of getting a blue ball = (p) = 1/4  = 0.25

Total experiments (N) = 30

Expected value = Number of experiments × Probability

 = N × p

 = 30 × 0.25

 = 7.50

Therefore, the expected value of getting blue ball is approximately 8.

Example 6.17

A fair die is thrown. Find out the expected value of its outcomes.

Solution

If the random variable X is the top face of a tossed, fair, six sided die, then the  probability mass function of X is

P_x (x)= 1/6, for x = 1,2,3,4,5 and 6

The average toss, that is, the expected value of X is

Therefore, the expected toss of a fair six sided die is 3.5.

Example 6.18

Suppose the probability mass function of the discrete random variable is

What is the value of E(3X + 2X²) ?

Solution

Example 6.19

Consider a random variable X with probability density function

Find E(X) and V(X).

Solution

We know that,

Example 6.20

If f (x) is defined by  ( x) = ke^{−2 x} , 0 ≤ x < ∞ is a density function. Determine the constant k and also find mean.

Solution

Example 6.21

The time to failure in thousands of hours of an important piece of electronic equipment used in a manufactured DVD player has the density function.

Find the expected life of the piece of equipment.

Solution:

Therefore, the expected life of the piece of equipment is 1/3  hrs (in thousands).

Example 6.22

A commuter train arrives punctually at a station every 25 minutes. Each morning, a commuter leaves his house and casually walks to the train station. Let X denote the amount of time, in minutes, that commuter waits for the train from the time he reaches the train station. It is known that the probability density function of X is

Obtain and interpret the expected value of the random variable X.

Solution:

Expected value of the random variable is

Therefore, the expected waiting time of the commuter is 12.5 minutes.

Example 6.23

Suppose the life in hours of a radio tube has the probability density function

Find the mean of the life of a radio tube.

Solution:

We know that, the expected random variable

Therefore, the mean life of a radio tube is 7,358 hours.

Example 6.24

The probability density function of a random variable X is

Find the value of k and also find mean and variance for the random variable.

Solution: