Please do by hand. Thanks in advance. b and c below are a part of the problem.
Please do by hand. Thanks in advance. b and c below are a part of the...
Please do by hand. Thanks in advance.
4. Let X1 - uniform(0,3) and X2 – uniform(0,2) and suppose that Xi and X2 are independent. Find the pdf of Y1 = X1 + X2. (Hint: First find the joint pdf of Yi and Y2 = X2.)
Please do by hand. Thanks in advance.
5. Let X1 and X2 have joint pdf f(x1, x2) = 4xı, for 0 < x < x2 < l; and 0 otherwise. Find the pdf of Y = X/X2. (Hint: First find the joint pdf of Y and Y2 = X1.)
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Suppose that Xi and X2 are independent random variables each having PDF: : otherwise (a) Use the transformation technique to find the joint PDF of Yi and Ya where Y-X1 and ½ = Xi +X2. (b) Using your answer to part (a), and the fact that o Vu(1-u) find and identify the distribution of Y2.
2. Let Xi exp(1) and X2 ~ variables with rate 1. Let: erp(1) be independent and identically-distributed exponential random (a) What is the cdf of X1? b) What is the joint pdf of (Xi, X2)? (c) What is the joint pdf of (Y, Z)? d) What is the marginal pdf of z?
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
Consider two random variables X and X2 with the joint pdf Nn.za) ={Orm ekewhere 1, o?r2 < 1 Let Y X,X2 and Y2X2 be a joint transformation of (Xi, X2) (a) Find the support of (Y.%) and sketch it. (b) Find the inverse transformation. (c) Compute the Jacobian of the inverse transformation (d) Compute the joint pdf of (Yi, Y2) (e) Derive the marginal pdf of Y? from the joint pdf of (y,,Y2).
3. A random variable X is said to have a Cauchy(α, β) distribution if and only if it has PDF function Now, suppose that Xi and X2 are independent Cauchy(0, 1) random variables, and let Y = X1 + X2. Use the transformation technique to find and identify the distribution of Y by first finding the joint distribution of Xi and Y. (Seahin 3 4
5. Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Yı = X1 + X2 and ½ = X1 + ßX2. You are not given the constant β but it is known that Cov(Yi,Y) = 0. Find (a) the density of Y2 (b) Cov(Xy½),
Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Y-Xi X2 and Y2- XiBX2. You are not given the constant B but it is known that Cov(Yi, Y2)-0. Find (a) the density of Y (b) Cov(X2, Y2)
Please do by hand. Thanks in advance.
8. Let X and Y be random variables with a bivariate normal distribution with parameters • px = 5 • Ox= 3 • My = 3 • Oy = 2 • p=.4 a) Find the expected value and variance of Z=4X-Y. b) Find the covariance of X and Z. c) Identify the distribution of Y. d) Identify the distribution of Y|X = 5.