2) Let Yİ,Ý,, ,y, be independent and identically distributed from the distribution with density where c > 0 is a con...
2. (15pts) Let X1, X2 be independent and identically distributed with Uniform(0,) density. (a) Is Y-X1 + X2 a sufficient statistic for θ? Hint: You need to find the conditional density of (X1, X2) given Y = X1 + X2. (b) Consider now S := max(X1, X2). 1s S a sufficient statistics for θ?
Let X and Y be independent and identically distributed with marginal probability density function f(a)- 0 otherwise, where 8>0 (a) [6 pts] Use the convolution formula to find the probability density function of X +Y. (b) [6 pts) Find the joint probability density function of U X+Y and V- X+Y
1. Let Yi,Y2, ,y, be independent and identically distributed N( 1,02) random variables. Show that, EVn P( Y where ) denotes the cumulative distribution function of standard normal You need to show both the equalities
5. Let X and Y be independent and identically distributed with marginal probability density function İf a> 0, otherwise, e-ea f(a)-( where >0 (a) [6 pts] Use the convolution formula to find the probability density function of X +Y (b) (6 pts) Find the joint probability density function of V= X + Y U=X+Y and 5. Let X and Y be independent and identically distributed with marginal probability density function İf a> 0, otherwise, e-ea f(a)-( where >0 (a) [6...
Problem 8: Suppose the Ý, , , Y, β are independent and identically distributed random variables in the interval (0,1) with individual densities where β 〉 0. Further suppose that β has marginal density f(β) 482 exp(-2β). Derive f(B|Y, Y). Identify the distributional family for B and describe its parameters. Problem 8: Suppose the Ý, , , Y, β are independent and identically distributed random variables in the interval (0,1) with individual densities where β 〉 0. Further suppose that...
Let Yı, ..., Yn be an independent and identically distribution sample from the distribution function f(y) = 3y?, for 0 Sy <1. (a) Show the sample mean, y converges in probability to some constant, c. Find c. (b) Find a function that converges in probability to log(C).
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...
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(H = 0) = q = 1-p. (Such random variables are called Bernoulli random variables.) a Find the moment-generating function for the Bernoulli random variable Y b Find the moment-generating function for W = Yit Ye+ … + . c What is the distribution of W? 1.
2. Suppose that {X1, ..., Xn} are independent and identically distributed random variables from a distribution with p.d.f. See-ox if x > 0 f(x) = 10 if x = 0 Let Y = min <i<n X;. Find the p.d.f. of Y.