Exercise 11.4: Mathematical Expectation

EXERCISE 11.4

1. For the random variable X with the given probability mass function as below, find the mean and variance.

2. Two balls are drawn in succession without replacement from an urn containing four red balls and three black balls. Let X be the possible outcomes drawing red balls. Find the probability mass function and mean for X.

3. If μ and σ 2 are the mean and variance of the discrete random variable X , and E ( X + 3) = 10 and E ( X + 3)2 = 116 , find μ and σ 2.

4. Four fair coins are tossed once. Find the probability mass function, mean and variance for number of heads occurred.

5. A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station. Let X denote the amount of time, in minutes, that the student waits for the train from the time he reaches the train station. It is known that the pdf of X is

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

6. The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function

Find the expected life of this electronic equipment.

7. The probability density function of the random variable X is given by

find the mean and variance of X .

8. A lottery with 600 tickets gives one prize of ₹200, four prizes of ₹100, and six prizes of ₹ 50. If the ticket costs is ₹ 2, find the expected winning amount of a ticket.

Answers:

(1) (i) 2.3 , 2.81   (ii) 1.67 , 0.56   (iii) 5/3 , 1/18     (iv) 2,4

(2) 

(3) 

(5) 15 minutes    (6) 1/3   (7) 1/2 , 1/8     (8) Loss ₹. 0.50