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6. Suppose C) ~ N (C), Ģ:: ru). Find the distribution of X|Y. Hint use the...
(1 point) If Y-Σǐn 1 X, and every Xi is i.i.d with distribution binomial(n, p), find...
(1 point) If Y-Σǐn 1 X, and every Xi is i.i.d with distribution binomial(n, p), find the MGF of Y M(t) What is the distribution of Y? Select all that apply. There may be more than one correct answer A. binomial (n, m * p) B. negative binomial(m * n,p) C. negative binomial(n,p) D. negative binomial(m, p) E. binomial(m,p) F. binomial(n,p) G. binomial(m * n,p) H. negative binomial(n,m* p) I. None of the above
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1),...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
Suppose that f(x, y) = cx, for 0 y x 2. (a) Find c. (b) Find P(x > 1 and Y < (c) Find the margina...
Suppose that f(x, y) = cx, for 0 y x 2. (a) Find c. (b) Find P(x > 1 and Y < (c) Find the marginal pdf of X. (d) Find the conditional pdf of Y given that X = x. (e) Find E[Y IX x (f) Find E[E[YX]]. (g) Find Cov(X, Y) (h) Are X and Y independent?
Suppose that f(x, y) = cx, for 0 y x 2. (a) Find c. (b) Find P(x > 1 and Y
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the...
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the sample mean, then approximate P(1.1< 1.2) for -100. 2. Consider a random sample XX from CDF F(a) 1-1/ for z [1, 0o) and zero otherwise. (a) Find the limiting distribution of XiI.n, the smallest order statistic. (b) Find the limiting distribution of XI (c) Find the limiting distribution of n In X1:m- 3. Suppose that X,,, are iid. N(0,o2). Find a function of T(x)x...
6. (10 points) Suppose X – Exp(1) and Y = -In(X) (a) Find the cumulative distribution...
6. (10 points) Suppose X – Exp(1) and Y = -In(X) (a) Find the cumulative distribution function of Y. (b) Find the probability density function of Y. (c) Let X1, X2,...,be i.i.d. Exp(1), and let Mk = max(X1,..., Xk) (Maximum of X1, ..., Xk). Find the probability density function of Mk (Hint: P(min(X1, X2, X3) > k) = P(X1 > k, X2 > k, X3 > k), how about max ?) (d) Show that as k- , the CDF of...
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p....
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillar...
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
3. Consider n i.i.d. r.v.s. X1, .Xn, where X, Bin(p). Show that the conditional PMF of...
3. Consider n i.i.d. r.v.s. X1, .Xn, where X, Bin(p). Show that the conditional PMF of (Xi, X2, -.. , X) given a number of successes, where a; E 10,1 に! is uniform
Problem III. (12 points) Consider the following probability distribution. X 0 6 P(X = x) 1/4...
Problem III. (12 points) Consider the following probability distribution. X 0 6 P(X = x) 1/4 1/4 1/4 1/4 1. (2 points) Find E(X). 2. (5 points) Find the sampling distribution of the sample mean X for samples of size = 2. n = 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1, ... , Xn) is proposed as @(X1, ... , Xn) = -XI(X; #0, and X: #6), п i=1 where...
2. Suppose X ~ N (μ,5). Find the asymptotic distribution of X(1-X) using A-methods. 3. Let...
2. Suppose X ~ N (μ,5). Find the asymptotic distribution of X(1-X) using A-methods. 3. Let X denote that the sample mean of a random sample of Xi,Xn from a distribution that has pdf Let Y,-VFi(x-1). Note that X = lari Xi- (a) Show that Mx(t) = (ca-tryM f(x) = e-z, x > 0. Find lim+oo My, (t)
(1 point) If Y-Σǐn 1 X, and every Xi is i.i.d with distribution binomial(n, p), find the MGF of Y M(t) What is the distribution of Y? Select all that apply. There may be more than one correct answer A. binomial (n, m * p) B. negative binomial(m * n,p) C. negative binomial(n,p) D. negative binomial(m, p) E. binomial(m,p) F. binomial(n,p) G. binomial(m * n,p) H. negative binomial(n,m* p) I. None of the above
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
Suppose that f(x, y) = cx, for 0 y x 2. (a) Find c. (b) Find P(x > 1 and Y < (c) Find the marginal pdf of X. (d) Find the conditional pdf of Y given that X = x. (e) Find E[Y IX x (f) Find E[E[YX]]. (g) Find Cov(X, Y) (h) Are X and Y independent?
Suppose that f(x, y) = cx, for 0 y x 2. (a) Find c. (b) Find P(x > 1 and Y
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the sample mean, then approximate P(1.1< 1.2) for -100. 2. Consider a random sample XX from CDF F(a) 1-1/ for z [1, 0o) and zero otherwise. (a) Find the limiting distribution of XiI.n, the smallest order statistic. (b) Find the limiting distribution of XI (c) Find the limiting distribution of n In X1:m- 3. Suppose that X,,, are iid. N(0,o2). Find a function of T(x)x...
6. (10 points) Suppose X – Exp(1) and Y = -In(X) (a) Find the cumulative distribution function of Y. (b) Find the probability density function of Y. (c) Let X1, X2,...,be i.i.d. Exp(1), and let Mk = max(X1,..., Xk) (Maximum of X1, ..., Xk). Find the probability density function of Mk (Hint: P(min(X1, X2, X3) > k) = P(X1 > k, X2 > k, X3 > k), how about max ?) (d) Show that as k- , the CDF of...
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
3. Consider n i.i.d. r.v.s. X1, .Xn, where X, Bin(p). Show that the conditional PMF of (Xi, X2, -.. , X) given a number of successes, where a; E 10,1 に! is uniform
Problem III. (12 points) Consider the following probability distribution. X 0 6 P(X = x) 1/4 1/4 1/4 1/4 1. (2 points) Find E(X). 2. (5 points) Find the sampling distribution of the sample mean X for samples of size = 2. n = 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1, ... , Xn) is proposed as @(X1, ... , Xn) = -XI(X; #0, and X: #6), п i=1 where...
2. Suppose X ~ N (μ,5). Find the asymptotic distribution of X(1-X) using A-methods. 3. Let X denote that the sample mean of a random sample of Xi,Xn from a distribution that has pdf Let Y,-VFi(x-1). Note that X = lari Xi- (a) Show that Mx(t) = (ca-tryM f(x) = e-z, x > 0. Find lim+oo My, (t)
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