Maths: Mathematical Expectation: Example Solved Problems with Answer, Solution, Formula

**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*(3*X*
+ 2*X*^{2}) ?

*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
≤

*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:*

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12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation : Mathematical Expectation: Solved Example Problems |

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