Can someone help me with this? Show that two jointly normally distributed random variables are independent if they are uncorrelated?
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Can someone help me with this? Show that two jointly normally distributed random variables are independent...
2. Suppose X and Y are independent continuous random variables. Show that P(Y < X) = | Fy(x) · fx (x) dx -oo where Fy is the CDF of Y and fx is the PDF of X [hint: P[Y E A] = S.P(Y E A|X = x) · fx(x) dx]. Rewrite the above equation as an expectation of a function of X, i.e. P(Y < X) = Ex[•]. Use the above relation to compute P[Y < X] if X~Exp (2)...
Exercise 2 (2). Let X and ε be independent normally distributed random variables such that X∼N(5,4)andε∼N(0,9).LetY bearandomvariablegivenbyY =1+2X+ε.Compute: (a) E(Y ) (b) Var(Y ) (c) Cov(X, Y ) (d) Corr(X, Y ) (e) What is the value of the ratio Cov(X, Y )/Var(X) ? (f) If Y = 1 + 3X + ε instead, what would be the value of Cov(X, Y )/Var(X) ? (g) If Y = 1 + 7X + ε instead, what would be the value of...
Let X1 and X2 be independent random variables with means μ1 and μ2, and variances σ21 and σ22, respectively. Find the correlation of X1 and X1 + X2. Note that: The covariance of random variables X; Y is dened by Cov(X; Y ) = E[(X - E(X))(Y - E(Y ))]. The correlation of X; Y is dened by Corr(X; Y ) =Cov(X; Y ) / √ Var(X)Var(Y )
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?
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
Let the frequency function of the joint distribution of the random variables X and Y P(X = 2, Y = 3) = P(X = 1, Y = 2) = P(X = -1, Y = 1) = P(X = 0, Y = -1) = P(X = -1, Y = -2) = 3 a) Determine the marginal distributions of the random variables X and Y. b) Determine Cov(X,Y) and Corr(X,Y). c) Determine the conditional distributions of the random variable Y as a...
(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22 0, for |2|<1, otherwise. fy(y) = for y>0, otherwise. 0, Let W = XY (a) (2 points) Find the p.d.f. of W, fw(w). (b) (2 points) Find the moment generating function of W2, Mw?(t) = E (e«w?). (c) (2 points) Find the conditional expectation of W given Y = y, E(W|Y = y).
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
2. A continuous random variable has joint pdf f(x, y): xy 0 x 1, 0sys 2 f(x, y) otherwise 0 a) Find c b) Find P(X Y 1) b) Find fx(x) and fy(v) c) Are X and Y independent? Justify your answer d) Find Cov(X, Y) and Corr(X, Y) e) Find fxiy (xly) and fyixylx)
Reserve Problems Chapter 5 Section 4 Problem 3 Suppose that X and Y are independent continuous random variables. Show that oxy o If X and Y are independant, then fxy (x, y) = – fx (x) - and the range of (X, Y) is rectangular. Therefore, fyy) / xyfx (x) dx E(X) fy(x) [fy(x) dx E(Y) Hence, Oxy = 0 Fan- | | C = 15 If X and Y are independant, then fxx (x, y) = fx (x) +fy...