ANSWER ALL QUESTIONS. BE SURE TO ANSWER

7.1.8. A balanced die is rolled 3 times. Let x1 be the number of times 1 appears and x2 be the number of times 2 appears. Work out the joint probability function of x1 and x2 .

7.1.9. A box contains 10 red, 12 green and 14 white identical marbles. Marbles are picked at random, one by one, without replacement. 8 such marbles are picked. Let x1 be the number of red marbles, x2 be the number of green marbles obtained out of these 8 marbles. Construct the joint probability function of x1 and x2

7.2.1. Compute (1) marginal probability functions; conditional probability functions of (2) x given y = 1; (3) y given x = 1, in Exercise 7.1.1 (1).

7.2.2. Compute the marginal and conditional probability functions, for all values of the conditioned variables, in Exercise 7.1.1 (2).

7.2.3. Construct the conditional density of y given x and the marginal density of x in Exercise 7.1.2 (1). What is the marginal density of y here?

7.2.4. Repeat Exercise 7.2.3 for the function in Exercise 7.1.2 (2), if possible. 7.2.5. Repeat Exercise 7.2.3 for the function in Exercise 7.1.2 (3), if possible.

7.4.1. Let x N(0, 1), a standard normal variable. Let y = a + bx + cx2 , c 0 be a quadratic function of x. Compute the correlation between x and y here and write it as a function of b and c. By selecting b and c show that, while a perfect mathematical relationship existing between x and y, as given above, can be made zero, very small ||, very large || (nearly 1 or 1, but not equal to 1). Thus it is meaningless to interpret relationship between x and y based on the magnitude or sign of .

7.4.2. By using Exercise 7.4.1 show that the following statements are incorrect: " > 0 means increasing values of x go with increasing values of y or decreasing values of x go with decreasing values of y"; " < 0 means the increasing values of x go with decreasing values of y or vice versa;" " near to 1 or 1 means near linearity between x and y".

7.4.3. Compute (1) covariance between x and y; (2) E[xy2 ]; (3) for the following discrete probability function: f(0, 1) = 1 5 , f(0, 1) = 2 5 , f(1, 1) = 1 5 , f(1, 1) = 1 5 and f(x, y) = 0 elsewhere.

7.4.2. By using Exercise 7.4.1 show that the following statements are incorrect: " > 0 means increasing values of x go with increasing values of y or decreasing values of x go with decreasing values of y"; " < 0 means the increasing values of x go with decreasing values of y or vice versa;" " near to 1 or 1 means near linearity between x and y".

7.4.4. Compute (1) (x, y); (2) E[x 3 y 2 ]; (3) for the following density function: f(x, y) = 1, 0 x 1, 0 y 1 and f(x, y) = 0 elsewhere. 7.4.5. Let x1 ,... , xn be a simple random sample of size n from a gamma population with parameters (, ). Let x= (x1 + + xn )/n. (1) Compute the moment generating function of x; (2) Show that n j=1 xj as well as xare gamma distributed. (3) Compute the moment generating function of u = xE[x] Var(x) . 7.4.6. (1) Show that u in Exercise

7.4.5 is a re-located, re-scaled gamma random variable for every n. (2) Show also that when n , u goes to a standard normal variable. 7.4.7. Going for an interview consists of t 1 = time taken for travel to the venue, t 2 = waiting to be called for interview and t3 = the actual interview time, thus the total time spent for the interview is t = t 1 + t 2 + t3 . It is known from previous experience that t 1 ,t 2 ,t3 are independently gamma distributed with scale parameter = 2 and E[t 1 ] = 6, E[t 2 ] = 8, E[t3 ] = 6. What is the distribution of t, work out its density.

. Let x1 , x2 be iid random variables from a uniform population over [0, 1]. Compute the following probabilities without computing the density of x1 + x2 ; (1) Pr{x1 + x2 1}; (2) Pr{x 2 3 }; (3) Pr{x 2 1 + x 2 2 1}. 7.4.9. If the real scalar variables x and y are independently distributed, are the following variables independently distributed? (1) u = ax and v = by where a and b are constants; (2) u = ax + b and v = cy + d where a, b, c, d are constants.

7.5.4. Construct an example of a joint density of x and y where E(y|x) = 1 + x + 2x 2 and (a) E(y) exists but E(y 2 ) does not exist; (b) E(y) does not exist.

7.5.5. Construct the regression function of x1 on x2 , x3 and show that it is free of the regressed variables x2 and x3 in the following joint density, why is it free of x2 and x3 ? f(x1 , x2 , x3 ) = 2x15x23x3 for 0 x1 , x2 , x3 < and zero elsewhere.

7.6.1. Let x given > 0 be a Poisson random variable with parameter . Let have a prior gamma distribution. Compute (1) the unconditional probability function of x; (2) the posterior density of given x = 3; (3) Bayes' estimate of . 7.6.2. Let x given b be generalized gamma with density of the form g1 (x|b) = cx1 e bx , x 0, > 0, > 0 and c is the normalizing constant. Let b have a gamma distribution. Then answer (1), (2), (3) of Exercise

7.6.3. Let x| N(, 1) and let N(0, 1). Answer (1), (2), (3) of Exercise 7.6.1.

7.6.4. Let x|a be uniformly distributed over [0, a]. Let a have a prior Pareto density c a 5 , 2 < a < 4 where c is the normalizing constant. Answer (1), (2), (3) of Exercise 7.6.1. 7.6.5. Let x|p be binomial with parameters (n = 10, p). Let p have a prior power function density f 2 (p) = cp5 , 0 < p < 0.7 where c is the normalizing constant.

7.7.1. Use transformation of variable technique to show that the density of u = x1 + x2 is the same as the one obtained by partial fraction technique in Example 7.20. 7.7.2. Verify that (7.35) is a density. 7.7.3. If x1 and x2 are independently distributed type-1 beta random variables with different parameters, then evaluate the densities of (1): u = x1 x2 ; (2): v = x1 x2 . 7.7.4. Evaluate the densities of u and v in Exercise

7.7.3 by using the following technique: Take the Mellin transform and then take the inverse Mellin transform to get the result. For example, the Mellin transform of the unknown density g(u) of u is available from E[u s1 ] = E[x s1 1 ]E[x s1 2 ] due to statistical independence and these individual expected values are available from the corresponding type-1 beta densities. Then take the inverse Mellin transform.

7.7.5. Let x1 and x2 be independently distributed gamma random variables with the parameters (1 , ) and (2 , ) with the same beta. By using transformation of variables, show that u = x1 x1+x2 is type-1 beta distributed, v = x1 x2 is type-2 beta distributed, w = x1 +x2 is gamma distributed. [Hint: Use the transformation x1 = r cos2 , x2 = r sin2 . Then J = 2r cos sin.]

8.2.1. In a factory, three machines are producing nuts of a certain diameter. These machines also sometimes produce defective nuts (nuts which do not satisfy quality specifications). Machine 1 is known to produce 40% of the defective nuts, machine 2, 30%, machine 3, 20% and machine 4, 10%. From a day's production, 5 nuts are selected at random and 3 are defective. What is the probability that one defective came from machine 1, and the other 2 from machine 2? 8.2.2. Cars on the roads in Kerala are known to be such that 40% are of Indian make, 30% of Indo-Japanese make and 30% others. Out of the 10 cars which came to a toll booth at a particular time, what is the probability that s are Indo-Japanese and 4 are Indian make?

8.2.3. A small township has households belonging to the following income groups, based on monthly incomes. (Income group, Number) = (<10 000, 100), (10 000 to 20 000, 50), (over 30 000, 50). Four families are selected from this township, at random. What is the probability that two are in the group (10 000 to 20 000) and two are in the group (<10 000)?

8.2.4. A class consists of students in the following age groups: (Age group, Number) = (below 20, 10), (20 to 21, 15), (21 to 22, 20), (above 22, 5). A set of four students is selected ar random. What is the probability that there are one each from each age group? 8.2.5. In Exercise 8.2.4, what is the probability that at least one group has none in the selected set?