Provide Information Regarding Statistics & Econometrics : Mathematical Expectation Lecture 22

Mathematical Expectation

Lecture 22

Let X1, X2,... Xn be the values of a random variable, and f(x1), f(x2),... f(xn) are their respective probabilities such that ∑f(x)=1. Then the expected value of X is denoted by E(X) and can be obtained as:

E(X) = X1f(x1) + X2f(x2) + ...+ Xnf(xn)

E(X)=∑X f(x)

In the case of a continuous random variable:

E(X)=∫xf(x)dx

E(X) is also called mean.

Mathematical Expectation of the Function of Random Variable

The function of a random variable (H(x)) is also a random variable. The mathematical expectation can be obtained in the same manner as in a random variable.

E(X)=∑H(x) f(x)

E(X)=∫ h(x) f(x)dx

H(x) may be X + 2, 2X -3, etc.

The variance can be obtained as:

Where:

Example 6.6: The probability distribution of a discrete random variable X is given by.

X	1	2	3
f(x)	2 / 7	3 / 7	2 / 7	1

Find the mean, variance, and standard deviation.

Solution:

E(X)=∑H(x) f(x)

E(X)= 14 / 7

E(X)= 2

Example 6.7: A dice is thrown. Find out the expected value of its outcomes.

Solution: The sample space of fair dice is given by:

S = {1, 2, 3, 4, 5, 6)

n (S) = 6

Let X represent the outcomes

The expected value of the outcomes of a dice is given by:

Example 6.8: If it is raining, an umbrella salesman can earn $30 per day. If it is fair, he can lose $6 per day. What is the expectation if the probability of rain is 0.3?

Solution: Let X1 represent earn per day & X2 represent lose per day.

X1 = $30 with probability f(x1) = 0.30 and X2 = - 6 with probability f(x2) = 0.70

E(X) = X1f(x1) + X2f(x2)

E(X) = 30 X 0.30 + (- 6) X 0.70

E(X) = $ 4.8

The average earning is $4.8.

Example 6.9: In a business firm, 5 men and 4 women apply for an executive post. An interview is scheduled for two of the candidates. Find the average number of women selected for interview.

Solution: Let X represent the number of women. First, it is needed to determine the probability for the number of women.