1. Consider sequence of independent identically distributed binary random variable x,,x,,x,,x,-4 ...
(3) Consider a sequence of independent and identically distributed random variables such that Xk-0, with common mean EĮXk] = 1. Define the Xi, X2, ,Xp, sequence k=1 (a) Compute E[ (b) Show that
(3) Consider a sequence of independent and identically distributed random variables such that Xk-0, with common mean EĮXk] = 1. Define the Xi, X2, ,Xp, sequence k=1 (a) Compute E[ (b) Show that
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
(10 points) Consider the infinite sequence of independent and identically distributed (stan- dard) uniform random variables: U1, U2, ..., i.e., Ui » Uniform(0,1). Also let N ~ Poisson(a). Assume N is independent of {U;}i>1. Consider the random variable z = į V. Calculate EZ. (Hint: use conditioning.)
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
4. Let X1,..., Xn be independent, identically distributed random vari- ables with common density 2 log c)? f(0; 1) = 0<<1, XCV21 (>0). : 212 (a) Find the form of the critical region C'* for the most powerful test of H:/= 1 vs. HQ: >1. (b) Suppose the n = 20 and a = .10. Find the specific value for the cutoff value) K from the critical region C* in part (a). (Hint: Show that Y = (log X/X) is...
(1 point) If X and Y are independent and identically distributed uniform random variables on (0, 1), compute each of the following joint densities U,v(u, v
(1 point) If X and Y are independent and identically distributed uniform random variables on (0, 1), compute each of the following joint densities. (a) U -3X, V - 3X/Y. fu.v(u, v) - (b) U - 5X + Y, V - 3X/(X + Y)
1. The random variables Xi, X2,.. are independent and identically distributed (iid), each with pdf f given in Assignment 4, Question 1. Let Sn- Xi+.+X Using the Central Limit Theorem and the graph of the standard normal distribution in Figure 1, approximate the probability P(S100 >600). Express your answer in the format x.x-10-x. Verify your answer by simulating 10,000 outcomes of Si00 and counting how many of them are > 600. Show the code 1.00 0.95 0.90 0.85 1.2 1.4...