(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with...
15. Let X,, X2,.. . be independent, identically distributed random variables, EIXI oo, and denote S,-X1+... + Xn. Prove that [Use symmetry in the final step.] 15. Let X,, X2,.. . be independent, identically distributed random variables, EIXI oo, and denote S,-X1+... + Xn. Prove that [Use symmetry in the final step.]
Let X1, , X2 ... be a sequence of independent and identically distributed continuous random variables. Say that a peak occurs at time n if Xn-1 < Xn < Xn+1 . Argue that the proportion of time that a peak occurs is, with probability 1, equal to 1/3
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
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).
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...
74. Let X1, X2, ... be a sequence of independent identically distributed contin- uous random variables. We say that a record occurs at time n if X > max(X1,..., Xn-1). That is, X, is a record if it is larger than each of X1, ... , Xn-1. Show (i) P{a record occurs at time n}=1/n; (ii) E[number of records by time n] = {}_1/i; (iii) Var(number of records by time n) = 2/_ (i - 1)/;2; (iv) Let N =...
Let X1 + X2 +...+ X30 be independent and identically distributed exponential random variables with mean 1. Calculate the probability that X ¯ is greater than 1.1. a. 29% b. 71% c. 35%
(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
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint probability density function: fX,Y (x,y) = e-y, 0 <x<y< ; =0 , elsewhere. Evaluate P( X2>X1, Y2>Y1) + P (X2 <X1, Y2<Y1) .
If X1 and X2 are independent and identically distributed normal random variables with mean m and variance s2, find the probability distribution function for U=X1-3X2/2.