L.11) Sums of independent random variables a) If X1 , X2 X, , , Xn are...
a) If X1 and X2 are independent random variables and X1 tollows the Nor nalLA σ1 X, +X2 follow? di tri t on and X to ows the Nonna μα 2 distribution, ne ha distribution do b) IfX1 , X2 . X, , arendependent random variables and each Xk follows the NormalA 에 ds rbutio. then what distribution does follow? , n L.6) Generating functions for sums of independent random variables a) If X and X are independent random variables,...
Let X1, X2, ....,. Xn, be a set of independent random variables, each distributed as a normal random variable with parameters μί and σ. Let х, ai Use properties of moment generating functions to determine the distribution of Y, meaning: find the type of distribution we get, and its expected value and variance
Let X1, X2, ..., Xr be independent exponential random variables with parameter λ. a. Find the moment-generating function of Y = X1 + X2 + ... + Xr. b. What is the distribution of the random variable Y?
Let X1, X2, . . . , Xn be a sequence of independent random variables, all having a common density function fX . Let A = Sn/n be their average. Find fA if (a) fX (x) = (1/ √ 2π)e −x 2/2 (normal density). (b) fX (x) = e −x (exponential density). Hint: Write fA(x) in terms of fSn (x).
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c) Sn=X1+X2 + . . . + Xn. (d) An -Sn/n 9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c)...
Let X1, X2, · · · be independent random variables, Xn ∼ U(−1/n, 1/n). Let X be a random variable with P(X = 0) = 1. (a) what is the CDF of Xn? (b) Does Xn converge to X in distribution? in probability?
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
3. Suppose that X1, X2, , Xn are independent random variables with the same expectation μ and the same variance σ2. Let X--ΣΑι Xi. Find the expectation and variance of
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
Let X1, X2,..., X, be n independent random variables sharing the same probability distribution with mean y and variance o? (> 1). Then, as n tends to infinity the distribution of the following random variable X1 + X2 + ... + x, nu vno converges to Select one: A. an exponential distribution B. a normal distribution with parameters hi and o? C a normal distribution with parameters 0 and 1 D. a Poisson distribution