please help me! Thanks in advance :)
please help me! Thanks in advance :) 5. Let N be a Poisson random variable with...
5. Let N be a Poisson random variable with parameter λ. Suppose ξι, ξ2, is a sequence of 1.1.d. random variables with mean μ and variance σ2, independent of N. Let SN ξι + ξΝ. Determine the mean and variance of SN.
Let N be a Poisson random variable with parameter λ. Suppose ξ1,ξ2,... is a sequence of i.i.d. random variables with mean µ and variance σ2, independent of N. Let SN = ξ1 + ...ξN. Determine the mean and variance of SN
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale. 3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
Let Xi and X2 be independent Poisson random variables with means λ! and λ2. (a) Find the distribution of x, + x, (b) Compute the conditional distribution of Xi given that X + X.-n
3, Let X be a Poisson random variable with parameter λ. Calculate the conditional expectation of X given that X is odd.
5. Let X be a Poisson random variable with parameter λ = 6, and let Y = min(X, 12). (a) What is the p.m.f. of X? (b) What is the mean of X? (c) What is the variance of X? (d) What is the p.m.f. of Y? (e) Compute EY
4. Let Xi,X2, , Xn be n i.id. exponential random variables with parameter λ > Let X(i) < X(2) < < X(n) be their order statistics. Define Yǐ = nX(1) and Ya = (n +1 - k)(Xh) Xk-n) for 1 < k Sn. Find the joint probability density function of y, . . . , h. Are they independent? 15In
33. Let X and Y be independent exponential random variables with respective rates λ and μ. (a) Argue that, conditional on X> Y, the random variables min(X, Y) and X -Y are independent. (b) Use part (a) to conclude that for any positive constant c E[min(X, Y)IX > Y + c] = E[min(X, Y)|X > Y] = E[min(X, Y)] = λ+p (c) Give a verbal explanation of why min(X, Y) and X - Y are (unconditionally) independent. 33. Let X...
Question 6 Let X1, . . . , Xn denote a sequence of independent and identically distributed i.id. N(14x, σ2) random variables, and let Yı, . . . , Yrn denote an independent sequence of iid. Nụy, σ2) ran- dom variables. il Λί and Y is an unbiased estimator of μ for any value of λ in the unit interval, i.e. 0 < λ < 1. 2. Verify that the variance of this estimator is minimised when and determine the...
Let X be a Poisson random variable with parameter λ = 6, and let Y = min(X, 12). (a) What is the p.m.f. of X? (b) What is the mean of X? (c) What is the variance of X? (d) What is the p.m.f. of Y? (e) Compute E[Y ].