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.

5.11. 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 . 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 Other commonly used discrete distributions | 131

(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?

6.1. Evaluate E(x) and E(x 2 ) for the uniform density by differentiating the moment generating function in (6.2). Unangemeldet Heruntergeladen am | 08.01.19 04:37 168 | 6 Commonly used density functions 6.2. Obtain E(x), E(x 2 ), thereby the variance of the gamma random variable by using the moment generating function M(t) in (6.14), (i) by expanding M(t); (ii) by differentiating M(t).

6.3. Expand the exponential part in the incomplete gamma integral (; a), integrate term by term and obtain the series as a 1F1 hypergeometric series. 6.4. Expand the factor (1 x) 1 in the incomplete beta integral in (6.27), integrate term by term and obtain a 2F1 hypergeometric series. 6.5. Show that the functions f 11(x) to f55(x) given in Section 6.11 are all densities, that is, show that the functions are non-negative and the total integral is 1 in each case. 6.6. For the functions in Exercise 6.5, compute (1) E(x), (2) Var(x); (3) the moment generating function of x, whenever these exist. 6.7. For the functions in Exercise 6.5 compute (1) the Mellin transform; (2) the Laplace transform wherever they exist and wherever the variable is positive. Give the conditions of existence.

. Let f(x) be a real-valued density function of the real random variable x. Let y be another real variable. Consider the functional equation f(x)f(y) = f(x 2 + y 2 ) where f is an arbitrary function. By solving this functional equation, show that f(x) is a Gaussian density with E(x) = = 0. 6.9. For the Exercise in 6.8, let z be a real variable and let the functional equation be f(x)f(y)f(z) = f(x 2 + y 2 + z 2 ). Show that the solution gives a Gaussian density with E(x) = = 0.

6.10. Shannon's entropy, which is a measure of uncertainty in a distribution and which has wide range of applications in many areas, especially in physics, is given by S = c [lnf(x)]f(x)dx where c is a constant and f(x) is a non-negative integrable function. Show that if S is maximized over all densities under the conditions (1) (i) f(x)dx = 1 then the resulting density is a uniform density; (2) Show that under the conditions (i) and (ii) E(x) = is given or fixed over all functional f then the resulting density is the exponential density; (3) Show that under the conditions (i), (ii) and (iii) E(x 2 ) = a given quantity, then the resulting density is a Gaussian density. [Hint: Use calculus of variations].