1. Suppose Z N(0, 1) ει ~ N(0, ơÐ €2 ~ N(0,03) independent and let (a) 12 pts] Under what conditi...
1.2 Let Yi and Y2 be independent random variables with Yi N(0, 1) and Y2 N(3,4). (a) What is the distribution of Y?? (b) If y-l (Y2-3)/2 | , obtain an expression for уту. What is its Yi and its distribution is yMVN(u, V), obtain an expression for yTV-ly. What is its distribution?
2. Let Z1 and Zo be independent standard normal random variables. Let! X= 221 +372 +12 X2 = 321 - 22 +11. (a) Find the joint density function of (X1, X2). (b) Find the covariance of X1 and X2. Now let Y1 = X1 + 4X2 +3 Y, = -2X2 +6X2 +5 (a) Find the joint density function of (Y1, Y). (b) Find the covariance of Yi and Y2.
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
Let X ~ N(0, 1), and let Z ~ Unif{-1, 1} (i.e. P(Z = -1) = P(Z = 1) = 1/2) be independent of X. Let Y = ZX. What is the distribution of Y? Show that X and Y are uncorrelated. Are X and Y independent?
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...
Problem 5 Suppose X and Y are independent random variables following Uniform[0, 1]. Let Z- (X +Y)/2. (1) Calculate the cumulative density of z. (2) Calculate the density of Z. Problem 5 Suppose X and Y are independent random variables following Uniform[0, 1]. Let Z- (X +Y)/2. (1) Calculate the cumulative density of z. (2) Calculate the density of Z.
1. Let {x, t,f 0) and {Yǐ.12 0) be independent Poisson processes,with rates λ and 2A, respectively. Obtain the conditionafdistributiono) Moreover, find EX Y X2t t given Yt-n, n = 1,2. 2, (a) Let T be an exponential random variable with parameter θ. For 12 0, compute (b) When Amelia walks from home to work, she has to cross the street at a certain point. Amelia needs a gap of a (units of time) in the traffic to cross the...
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1 with probability 1/2. Show that SZ N(0,1) 5. Let Z N(0, 1) and X = Z2. This distribution is called chi-square with degree of freedom. Calculate P(1 < X < 4) one N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1...
1. [30 pts! Let Yǐ follow a moving average process of order 1 (ie, MA(1): where e is a white noise process with N(0,1). Suppose that you estimate the model using STATA. You obtain ê-1, ê-0.5 and ớ2-1. You also know e,-2 and E1-1-3. (a) Obtain the unconditional mean and variance of Y (b) Obtain Cor(Y, Yi-1). (c) Obtain the autocorrelation of order 1 for Y 1. [30 pts! Let Yǐ follow a moving average process of order 1 (ie,...
1. [26 pts Let Uı, , Un be independent, identically distributed Unifomn random variables with (continu- ous) support on (0, b), where b> 0 is a parameter. (a) Define the random variable Y :--Σί 1 log(U,), where log is the natural logarithm function. De- termine the probability density function (pdf) p(y; b) ofY by explicitly computing it (b) Based on the pdf you found in part (a) above, determine the third moment of Y, i.e., EY] (c) Suppose now that...