Let X1 , X, , and X3 be independent and uniformly distributed between-2 and 2. (a) Find the CDF and PDF ofYX, +2X2 (b) Find the CDF of Z-), + X, . (c) Find the joint PDF of Y and Z.(: Try the tri...
Let X1 , X2 , and X3 be independent and uniformly distributed between -2 and 2. (a) Find the CDF and PDF of Y =X1 + 2X2 . (b) Find the CDF of Z = Y + X3 . (c) Find the joint PDF of Y and Z . (Hint: Try the trick in Problem 2(b))
Show the random variables X and Y are independent, or not independent Find the joint cdf given the joint pdf below Suppose that (X, Y) is uniformly distributed over the region defined by 0 sys1-x2 and -1sx 4 Therefore, the joint probability density function is, 0; Otherwise Suppose that (X, Y) is uniformly distributed over the region defined by 0 sys1-x2 and -1sx 4 Therefore, the joint probability density function is, 0; Otherwise
Problem 5 Let X and Y be random variables with joint PDF Px.y. Let ZX2Y2 and tan-1 (Y/X). Θ i. Find the joint PDF of Z and Θ in terms of the joint PDF of X and Y ii. Find the joint PDF of Z and Θ if X and Y are independent standard normal random variables. What kind of random variables are Z and Θ? Are they independent? Problem 5 Let X and Y be random variables with joint...
Suppose X1, X2, Xz~exp(1) and they are independent. (a) Compute the cdf of X1 (b) Let Y- max(Xi, X2, X3). Find the cdf of Y (c) Derive the pdf of Y
7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y = max(X,Xy) and Z-X1 + X2. (1) Calculate the joint PDF of Y and Z. (2) Derive the marginal PDF of both Y and Z. Are Y and Z independent? 7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y...
24. The joint cdf of (X,Y) is Find a) Joint pdf of (X, Y) b) Marginal pdf of X and Y c) PI(X s 1) n (Y s 1) d) PI(1 < X <3) n (1 <Y <2)] Page 4 of5
(a) Consider four independent rolls of a 6-sided die. Let X be the number of l's and let y be the number of 2's obtained. What is the joint PMF of X and Y? (b) Let X1, X2, X3 be independent random variables, uniformly distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is the middle of the three values). Find the conditional CDF of X1, given that Y = 0.5. Under this conditional distribution, is...
7. Let X and Y have joint pdf 122 (1-x)y, 0, 0〈x〈1,0くyく1. otherwise. x,y(x,y) = (a) Find the joint cdf of X and Y. (4pts) (b) Find PY< VX. (Spts) (c) Find the marginal pdfs of X and Y. (6pts) (d) Are X and Y independent? (5pts) 7. Let X and Y have joint pdf 122 (1-x)y, 0, 0〈x〈1,0くyく1. otherwise. x,y(x,y) = (a) Find the joint cdf of X and Y. (4pts) (b) Find PY
1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a. Find the joint pdf fxy(x, y), marginal pdf (fx(x) and fy()) and cdf (Fx(x) and Fy)) b. Find the conditional pdf fxiy Cr ly c. Find the probability P(X < Y = y) d. Are X and Y independent?
(5) 5. If X is uniformly distributed on [ - 1,1], find the cdf and pdf of Y = X2.