Let X ~ Bern(p). Find the distribution of Xk , k = 2, 3,...
Let Xi,..., Xn be iid random variables with distribution Bern(p) (a) Is the statistic 름 Σ. ? (b) Is the statistic (Σ¡X 2? Xi an unbiased estimator of p i) an unbiased estimator of p
Let Xi,..., Xn be iid random variables with distribution Bern(p) (a) Is the statistic 름 Σ. ? (b) Is the statistic (Σ¡X 2? Xi an unbiased estimator of p i) an unbiased estimator of p
2. Suppose X - Unif (0, 1) and S, |X ~ Bin(n, X). Let I, indicate the ith trial is a success. This 10, find: implies that llx ~iid Bern(p a) P(S1o 3) X). For n c) P(I11 1S10 3) d) P(l111, 12 1S10 3)
2. Suppose X - Unif (0, 1) and S, |X ~ Bin(n, X). Let I, indicate the ith trial is a success. This 10, find: implies that llx ~iid Bern(p a) P(S1o 3) X). For...
3. Let X be a random variable from a geometric distribution with parameter p (P(X- k p(1-P)"-, } k-1 k-1, 2, ...). Find Emin{X, 100
Suppose X = Exp(1) and Y= -ln(x)
(a)Find the cumulative distribution function of Y .
(b) Find the probability density function of Y .
(c) Let X1, X2, ... , Xk 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 >= kq, how about max ?)
(d) Show that as k → 00, the CDF...
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
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...
The random vector x (XI, X2,... ,Xk)' is said to have a symmetric multivariate normal distribution if x ~ Ne(μ, Σ) where μ 1k, i.e., the mean of each X, is equal to the same constant μ, and Σ is the equicorre- lation dispersion matrix, i.e. when k 3, μ-0, σ2-2 and ρ 1/2, find the probability that Hint: Recall that if x = (Xi, , Xk), has a continuous symmetric dis tribution, then all possible permutations of X1,... ,Xk...
Let x = (X1, . . . , Xk) ~ Mk(μ, Σ), with r(Σ) = k. (a) Show that (2が/2E11/2 = (b) Evaluate looo . . . Jo oo exp(-(z? + 2x1x2 + 4墎dzda2.
need the solution for this question.tq
Let X,,. X, be a random sample from a Poisson (a) (a). 2. distribution. Find the sufficient statistic for A. (25 marks) Let X,X,X, be a random sample from a gamma (k, B) (b). P.1 distribution with k is fixed. DefineX X, n피 based upon unbiased ness, consistency Evaluate (0). and efficiency is a minimum variance unbiased estimator for B Show that (ii). (75 marks) (2)3
Let X,,. X, be a random sample from...
A random variable X has the following mgf
et
M(t)=1−t, t<1.
(a) Find the value of ∞ (−1)k E(Xk).
(b) Find the value of E(2−X).
(c) Find the value of Var(2−X).
(d) Find the probability P (X > 4).
10. A random variable X has the following mgf М() t 1 1 t (a) Find the value of 1E(Xk) (b) Find the value of E(2X). (c) Find the value of Var(2-X) k 0 k! (d) Find the probability P(X >...