2. Consider the negative binomial distribution with parameters r,p and having pmf nb(x;r,p) = Ꮖ (* + r − ¹) p*(1 − p)² p'(1-p) x = 0, 1, 2, 3, … …. (a) Supposer 2, then show that T-1 p = X+r−1 is an unbiased estimator for p. (Hint: write out E(p), then cancel out x+r −1 inside the sum). (b) A reporter wishing to interview five individuals who support a certain candidate (for presidency?) begins asking people whether they support (S) or not support (F) the can- didate. If they observe the following sequence of responses SFFSfffffffffffSSS, esti- mate p the true proportion of people who support the candidate. How does the estimate change if the following sequence of responses were observed ssssfffffffffffffs. Does it matter to the estimate when the first four S's appear in the sequence of responses?

7. Let X be a continuous random variable with pdf f(x)=√ 0

2. Suppose the distribution of Y given X = x is a normal random variable with mean x and variance x². And X follows an uniform distribution on (0,1) (a) Find E(XY) (b) Let U = X, V = Y/X. Show that U and V are independent

Let Y, represent the ith normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Ya, Ya, Yin, of size n,, from the ith population with sample mean Y, and sample variance S?=₁1(Y-₁². (h) Find the standard error of U₂ in part (g), assuming that of = 0² = 0². (i) Discuss how the distribution of Y₁ - Y₂ can be used to test the equality of the two population means, #₁1 and 12, when o=o=o² is known. (i) Define appropriate rejection regions, in terms of Y₁-Y₂, for testing Ho: #₁ = 1₂ against a two-sided alternative hypothesis at the a level of significance.

3 For an SRS {4₁};=, from the normal distribution. ==// (Y-M) 2 fy (y;u, 0²) - 1 е värvoze Show that the sample mean I is a consistent estimator,

10. True or false. (i) –1 < Cov(X, Y) <1 for any two random variables X, Y. (ii) When two random variables have zero correlation they must be independent. (iii) Cov(13.X, 13X) = 13Var(X). Those statements that you mark as false, explain why and give a corrected version.

12. Let random variable X has geometric distribution with p.d.f. a a f(x) = pq*", k =1,2,3,... Show that E[X]=- and Var(X)= 4 without using its moment generating function.

29. Suppose X1, X2, X3 are a random sample of size 3 from a distribution with pdf f(x) = (1/0)ex/0 for x > 0, 0 > 0. Let ô₁ = X1, 02 = (X1+ X2)/2 and 3 = (X1+2X2)/3. Show that these estimators of 0 are all unbiased, and determine the relative efficiencies between them.

Suppose X, Y, Z are i.i.d. from a Normal distribution with mean μ and variance 4. You have to compare between three estimators for μ, which are    T1:= (2X + Y)/3 , T2:= (X + Y − Z)/2 , T3:= (Y + 2Z)/2 .   (a) Which among the above are unbiased estimators for μ? (b) Which among these have the smallest variance? (c) Can you propose an estimator for μ which is better than the 3 above? You have to show that your estimator is indeed better.

Let Y, represent the ith normal population with unknown mean #, and unknown variance of for i=1,2. Consider independent random samples, Ya,Ya. Yin, of size n,, from the ith population with sample mean Y, and sample variance S?=₁1(Yu - Y.². (g) For non-zero constants a,'s, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of = o2 = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #, and p2, when of=o=o² is known. (j) Define appropriate rejection regions, in terms of Y₁ - ₂, for testing Ho: #₁ = ₂ against a two-sided alternative hypothesis at the a level of significance.