c. Let X~Ber(p), i.e.
, ? = 0,1. Derive the variance V(X).
![Xu Ber(P) Pala)- pp (1-P) a=0,1 VOX)= ? We know that v(a)= E(X?)- [E(x)] n E (X) = La P() al o (p? (1-P))+1(PCI-P)) - 0(1-](http://img.homeworklib.com/questions/590ccf10-0dab-11eb-a2d4-51bcf0195609.png?x-oss-process=image/resize,w_560)
c. Let X~Ber(p), i.e. , ? = 0,1. Derive the variance V(X). px (I) = p'(1...
Let x1, x2,..,xn represent a
random sample from a distribution with pdf
f(x)=px(1-p)1-x for x=0,1 and 0<p<1.
Find MLE for p.
Choose an answer:
n O b. 1/29=1*; O d. None are correct 59
9.) Suppose that X is a continuous random variable with density C(1- if [0,1] px(x) ¡f x < 0 or x > 1. (a) Find C so that px is a probability density function. (b) Find the cumulative distribution of X (c) Calculate the probability that X є (0.1,0.9). (d) Calculate the mean and the variance of X
Let X1, X2,· · ·iid B(1, x), i.e,P(X1= 1) =x= 1−P(X1= 0), where x∈ [0,1]. Let Sn = X1+X2+· · ·+Xn. What can you say about the limiting behaviour of Sn/n from strong law large number
2. Let X ~ Bin(4, ), i.e., the PMF of X is given by Px(1) = (1) (1) *,* = 0,1,2,3,4. Find Px B(k) where B = {X #0}.
8. Let X,, X,..X be iid. rv.'s from P(2) (i.e. f(r)-, x-0,1.) Find the MLE fore 192n
Let X, Y, and Z be three i.i.d. geometric random variables with parameter p, i.e., the probability mass function of X is PX(k) = (1 − p) k−1p, k = 1, 2, . . . Find the conditional probability distribution of X + Y given X + Y + Z, i.e., find: P(X + Y = i|X + Y + Z = j).
a) Let X-Unif(0,1). Derive the pdf of Y =-ln(1-X) Remember to provide its support. Let X-N(1,02). Derive the pdf of Y-ex and remember to provide its support. b) Hint for both parts: First work out the cdf of Y, and then use it to find the density of Y.
8. Let X.(i-12) be independent N(0,1) random variables. a. Find the value of c such that P ( (X1 + X2尸/( X2-X)2 < c ) =.90 b. Find P(2 X1 -3 X21.5) c. Find 95th percentile of the distribution of Y-2X -3X2
Let X1, X2, ..., X, be a r.s. from P(X), f(x) = (a) Show that X1 is the unbiased estimator for 1. (b) Find îmle for X. (c) Derive the Fisher information I(). (d) Show that Amle is the unbiased estimator (UE) for 1 and Var(AMLE) attains CRLB(= mas). i.e., İmle is minimum variance unbiased estimator (MVUE).
Topology
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup norm of C[o, 1]. (i) Show that 5 is closed under pointwise multiplication, that is,if f,0€万 then fg P and, moreover, llfglloo for all f,g E P
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup...