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Unit 10: Approximate Expressions for Expectations and Variance

The retailer of these sets gets a commission @ 20%, 12%, 25% and 15% on the respective Notes
sets. What is the expected commission of the retailer?
Hint : Take commission as random variable.
3. Three cards are drawn at random successively, with replacement, from a well shuffled
pack of cards. Getting a card of diamond is termed as a success. Tabulate the probability
distribution of the number successes (X). Find the mean and variance of X.
Hint : The random variable takes values 0, 1, 2 and 3.
4. A discrete random variable can take all possible integral values from 1 to k each with
1
probability . Find the mean and variance of the distribution.
k

1 1 é ( k k + 1 )(2k + ) 1 ù
2
2
2
Hint : ( ) = (1 + 2 + .... k 2 ) = ê ú .
+
E X
k k ë 6 û
5. An insurance company charges, from a man aged 50, an annual premium of Rs 15 on a
policy of Rs 1,000. If the death rate is 6 per thousand per year for this age group, what is the
expected gain for the insurance company?
Hint : Random variable takes values 15 and - 985.
6. On buying a ticket, a player is allowed to toss three fair coins. He is paid number of rupees
equal to the number of heads appearing. What is the maximum amount the player should
be willing to pay for the ticket.
Hint : The maximum amount is equal to expected value.
7. The following is the probability distribution of the monthly demand of calculators :

Demand ( ) : 15 16 17 18 19 20
x
x
Probability p ( ) : 0.10 0.15 0.35 0.25 0.08 0.07
Calculate the expected demand for calculators. If the cost c of producing x calculators is
2
given by the relation c = 4x - 15x + 200, find expected cost.
Hint : See example 12.

Answers: Self Assessment

1. (b) 2. (a) 3. (d) 4. (c)

10.10 Further Readings

Books Introductory Probability and Statistical Applications by P.L. Meyer
Introduction to Mathematical Statistics by Hogg and Craig

Fundamentals of Mathematical Statistics by S.C. Gupta and V.K. Kapoor

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