5.1. Compute E(x 2 ) for the geometric probability law by summing up or by using the definition, that is, by evaluating E(x 2 ) = x=1 x 2q x1p.

5.2. Compute (i) E(x); (ii) E(x 2 ); for the negative binomial probability law by using the definition (by summing up).

5.3. Compute (i) E(x); (ii) E(x 2 ); by using the technique used in the geometric probability law by differentiating the negative binomial probability law.

5.4. Compute E(x) and E(x 2 ) by differentiating the moment generating function in the Poisson probability case.

5.5. Compute E(x) and variance of x by using the moment generating function in the binomial probability law.

5.6. Construct two examples of discrete probability functions where E(x) = Var(x).

5.7. Solve the difference-differential equation in (5.16) and show that the solution is the probability function given therein.

5.8. Show that the functions f7 (x) to f33(x) in Section 5.8 are all probability functions, that is, the functions are non-negative and the sum in each case is 1.

5.9. For the probability functions in Exercise 5.8, evaluate the first two moments about the origin, that is, E(x) and E(x 2 ), whenever they exist

5.10. Compute the truncation constant c so that cf(x) is a truncated probability function of f(x) in the following cases:

(i) Binomial probability function, truncated below x = 1 (Here, c = 1 c0 where c0 is given above);

(ii) Binomial probability, truncated at x = n;

(iii) Poisson probability function, truncated below x = 1;

(iv) Poisson probability function, truncated below x = 2;

(v) Geometric probability function, truncated below x = 2;

(vi) Geometric probability function, truncated above x = 10.

5.11. Compute the

(a) the probability generating function P(t),

(b) E(x) by using P(t), (c) E(x 2 ) by using P(t) for the following cases:

(i) Geometric probability law;

(ii) Negative binomial probability law.

5.12. A gambler is betting on a dice game. Two dice will be rolled once. The gambler puts in Rs 5 (His bet is Rs 5). If the same numbers turn up on the two dice, then the gambler wins double his bet, that is, Rs 10, otherwise he loses his bet (Rs 5). Assuming that the dice are balanced

(i) What is the gambling house's expected return per game from this gambler?

(ii) What is the probability of the gambler winning exactly five out of 10 such games?

(iii) What is the gambler's expected return in 10 such games?

5.13. Cars are arriving at a service station at the rate of 0.1 per minute, time being measured in minutes. Assuming a Poisson arrival of cars to this service station, what is the probability that (a) in a randomly selected twenty minute interval there are

(i) exactly 3 arrivals;

(ii) at least 2 arrivals;

(iii) no arrivals;

(b) if 5 such 20-minute intervals are selected at random then what is the probability that in at least one of these intervals (

i) (a)

(i) happens;

(ii) (a)

(ii) happens;

(iii) (a)

(iii) happens.

5.14. The number of floods in a local river during rainy season is known to follow a Poisson distribution with the expected number of floods 3. What is the probability that

(a) during one rainy season

(i) there are exactly 5 floods;

(ii) there is no flood;

(iii) at least one flood;

(b) if 3 rainy seasons are selected at random, then none of the seasons has

(i) (a)(i) happening;

(ii) (a)(ii) happening;

(iii) (a)(iii) happening;

(c) (i) (a)(i) happens for the first time at the 3rd season; (ii) (a)(iii) happens for the second time at the 3rd season.

5.15. From a well-shuffled deck of 52 playing cards (13 spades, 13 clubs, 13 hearts, 13 diamonds) a hand of 8 cards is selected at random. What is the probability that the hand contains

(i) 5 spades?

(ii) no spades?

(iii) 5 spades and 3 hearts?

(iv) 3 spades 2 clubs, 2 hearts, 1 diamond?