(x-2) 5. a) Let S Prove that s? Po? n-1 b) Consider a sequence of random...
Suppose the random variable X has probability density function (pdf) - { -1 < x<1 otherwise C fx (x) C0 : where c is a constant. (a) Show that c = 1/7; (b) Graph fx (х); (c) Given that all of the moments exist, why are all the odd moments of X zero? (d) What is the median of the distribution of X? (e) Find E (X2) and hence var X; (f) Let X1, fx (x) What is the limiting...
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0. Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and y1 = 8 and for n = 1,2,3,··· x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y . nn nn (a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all positive integers n. (xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive integers n. Hence, prove...
1. Let X1, X2, , Xn be independent Normal μ, σ2) random variables. Let y,-n Σ_lx, denote a sequence of random variables (a) Find E(y,) and Var(y,) for all n in terms of μ and σ2. (b) Find the PDF for Yn for alln. (c) Find the MGF for Yn for all n.
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
6. Let X1,..., Xn be a random sample from Uniform (0, 1). a) Find the exact distribution of U = – log(X(1)) where X(1) = min(X1, X2,..., Xn). b) Find the limiting distribution of n(1 – X(n)), where X(n) = max(X1, X2, ..., Xn).
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...
Let {Xn} be a sequence of iid random variables 1. (20 points) Let {Xn} be a sequence of iid random variables with common pdf f(x) = - =e-x2/2,x ER. Then find the limit in probability of the sequence of random variables {Y} where Yo: 31x11. i=1
Let X1, ..., Xn be a random sample from a population with pdf f(x 1/8,0 < x < θ, zero elsewhere. Let Yi < < Y, be the order statistics. Show that Y/Yn and Yn are independent random variables
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)? 8. Let X1,...,Xn denote a random...