ATTEMPT ALL QUESTIONS. GIVE DETAILED EXPLANATION

8.3.1. Evaluate the normalizing constant c1 in (8.16). Then evaluate the joint marginal densities of (x1 ,... , xk?1 ),

8.3.2. For the model in (8.17) evaluate E[x h1 1 ?x hk k ].

8.3.3. By using Exercise 8.3.2, or otherwise, show that x1 in the model (8.17) can be written equivalently as a product of independently distributed type-1 beta random variables. (Hint: Take E(x h 1 ) and look at the decomposition of this gamma product.)

8.3.4. Evaluate the normalizing constant c2 in (8.17).

8.3.5. Evaluate the normalizing constant c3 in (8.18).

8.3.6. Take the sum u = x1 +? +xk , the sum of type-1 Dirichlet variables. In Result 8.2, it is shown that u is type-1 beta variable. By using the fact that if u is type-1 beta, then u 1?u and 1 1?u are type-2 beta variables write down the results on

8.3.7. It is shown in Result 8.4 that u = 1 1+x1+?+xk is a type-1 beta if x1 ,... , xk have a type-2 Dirichlet distribution. Using the fact that if u is type-1 beta, then u 1?u and 1 1?u are type-2 beta distributed, write down the corresponding results

8.3.8. Using Exercises 8.3.6 and 8.3.7 and by using the properties that if w is type-2 beta, then 1 w is type-2 beta, 1 1+w is type-1 beta, w 1+w is type-1 beta write down the corresponding results on when x1 ,... , xk have a type-2 Dirichlet distribution. 8.3.9. If (x1 ,... , xk ) is type-1 Dirichlet, then evaluate the conditional density of x1 given x2

8.3.10. For k = 2, consider type-1 and type-2 Dirichlet densities. By using Maple or Mathematica, draw the 3-dimensional surfaces for (1) fixed ?1 , ?2 and varying ?3 ; (2) fixed ?2 , ?3 and varying ?1

8.4.2. In Exercise 8.4.1, evaluate the conditional density of X(1) given X(2) and show that the it is also a r-variate Gaussian. Evaluate (1) E[X(1) |X(2) ], (2) covariance matrix of X(1) given X(2) .

8.4.3. Answer the questions in Exercise 8.4.2 if r = 1, p ? r = p ? 1.

8.4.4. Show that when m = 1 the matrix-variate Gaussian becomes n-variate normal. What are the mean value and covariance matrix in this case?

8.4.5. Write the explicit form of a p-variate normal density for p = 2. Compute (1) the mean value vector; (2) the covariance matrix; (3: correlation ? between the two components and show that ?1

8.5.1. Evaluate the integral ? X>0 e ?tr(X)dX and write down the conditions needed for the convergence of the integral, where the matrix is p p. 8.5.2. Starting with the integral representation of ?p (?) and then taking the product ?p (?)?p (?) and treating it as a double integral

9.3.1. Use a computer and select random numbers between 0 and 1. This is equivalent to taking independent observations from a uniform population over [0, 1]. For each point, starting from the number of points n = 5, calculate the standardized sample mean z = ?n(x???) ? , remembering that for a uniform random variable over [0, 1], ? = 1 2 , ? 2 = 1 12 . Make many samples of size 5, form the frequency table of z values and smooth to get the approximate curve. Repeat this for samples of sizes, n = 5, 6,... and estimate n so that the simulated curve approximates well with a standard normal curve.

9.3.2. Repeat Exercise 9.3.1 if the population is exponential with mean value ? = 5. [Select a random number from the interval [0, 1]. Convert that into an observation from the exponential population by the probability integral transformation of Section 6.8 in Chapter 6, and then proceed.]

9.3.3. Consider the standardized sample mean when the sample comes from a gamma population with the scale parameter ? = 1 and shape parameter ? = 5. Show that the standardized sample mean is a relocated and re-scaled gamma variable.

9.3.4. By using a computer or with the help of MAPLE or MATHEMATICA, compute the upper 5% tail as a function of n, the sample size. Determine n when the upper tail has good agreement with the upper 5% tail from a standard normal variable. 9.3.5. Repeat the same Exercise

9.3.4 when the population is Bernoulli with the probability of success (1) p = 1 2 , symmetric case; (2) p = 0.2 non-symmetric case.

10.2.1. If x1 ,... , xn are iid from a uniform population over [0, 1], evaluate the density of x1 + ? + xn for (1) n = 2; (2) n = 3. What is the distribution in the general case?

10.2.2. If x1 ,... , xn are iid Poisson distributed with parameter ?, then (1) derive the probability function

7.1.1. Check whether the following are probability functions: (1) f(D)=. f(-1,2) = 7, f (-1,3) = = f(0, 1) =: f(0,2) = 20 f(0, 3) = 8 and f(x,y) =0, elsewhere. (2) f(-2,0) =1 and f(x,y) =0, elsewhere. (3) f3.1)= 3, f(3,2)=- FOOD)= f(0,2) = and f(x,y) =0, elsewhere. 7.1.2. Check whether the following are density functions: (1) f(x,y) = -00 0, j= 1,....k, x, = 0, 1,...,n, j= 1,...,k-1 and g2(Pp>--- .Pk-1/ X1...*_1) =0 elsewhere. These density functions (1) and (2) are very important in Bayesian analysis and Bayesian statistical inference