Problem 2. (5 marks. 3, 2) Let Yi and Y2 be two independent discrete random variables...
Let Y1 and Y2 be two independent discrete random variables such that: p1(y1) = 1/3; y1 = -2 ,- 1, 0 p2(y2) = 1/2; y2 = 1, 6 Let K = Y1 + Y2 a) FInd the moment Generating function of Y1, Y2, and K b) find the probability mass function of K
Problem 2. (3 marks; 3, 2) Let Yi and Y, be two independeet discrete Let K-Y+Y a) Find the moment generating function of Y.Ys and K. For bistre ee.t) Le 64 24 Page 6 of dedicmail.utoronto.ca b) Using part a), find the probability mass function of K 64 know mklt)- 24
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.
0.4. Suppose that Yi and Y2 are discrete independent random variables with the following moment generating functions: 6 10 102 I. Find the mean and variance of Yi 1- 0.4. Suppose that Yi and Y2 are discrete independent random variables with the following moment generating functions: 6 10 102 I. Find the mean and variance of Yi 1-
0.4. Suppose that Yi and Y2 are discrete independent random variables with the following moment generating functions: 6 10 102 I. Find the mean and variance of Yi 1-
2. Let the random variables Y1 and Y, have joint density Ayſy22 - y2) 0<yi <1, 0 < y2 < 2 f(y1, y2) = { otherwise Stom.vn) = { isiml2 –») 05451,05 ms one a independent, amits your respon a) Are Y1 and Y2 independent? Justify your response. b) Find P(Y1Y2 < 0.5). on the
Consider two binomial random variables Yi and Y2. In particular Y1 binom(n, Pi) Y2 ~binom(n, p2) Propose and justify a consistent estimator for pi -p2
Consider k groups and considerYi,Y2...,Yk independent random variables, where Yi ~b(ni,P)1,2,.., k, Let ni be a fixed value representing the number of items and Pi the probability that they respond to a fixed non-random covariate represented by di, di, d, d. Then we have, In 1-pi with parameters /o and A. Find a two-dimensional sufficient statistic for (Bo,B)
3. Suppose that Yi and 2 are continuous random variables with joint pdf given by and zero otherwise, for some constant c >。 (a) Find the value of c. (b) Are Yi and Y2 independent ? Justify your answer. (c) Let Y = Yi + ½. compute the probability P(Y 3). (d) Let U and V be independent continuous random variables having the same (marginal) distri- 3 MARKS 1 MARK 3 MARKS bution as Y2. Identify the distribution of random...
Let Y1, Y2, ..., Yn be independent random variables each having uniform distribution on the interval (0, θ). Find variance(Y(j) − Y(i)) Let Yİ,Y2, , Yn be independent random variables each having uniform distribu - tion on the interval (0,0) Fin ar(Y)-Yo