Suppose X ~N(0,9) and Y ~N(1,16) X and Y are independent, then P(0<X+Y<2) =
TOPIC: Additive property of independent Normal variates and finding the required probability.
Suppose X ~N(0,9) and Y ~N(1,16) X and Y are independent, then P(0<X+Y<2) =
Suppose X, Y are independent and X~N(1,4) and Y N(1,9). If P(2X Y a) P(4X - 2Y 2 4a), then find a
is independent of X, and e Problem 3 Suppose X N(0, 1 -2) -1 <p< 1. (1) Explain that the conditional distribution [Y|X = x] ~N(px, 1 - p2) (2) Calculate the joint density f(x, y) (3) Calculate E(Y) and Var(Y) (4) Calculate Cov(X, Y) N(0, 1), and Y = pX + €, where
2. Suppose that X1, ..., Xd N(0,9) and suppose that Y1, ..., Y10 d (1,9). Assume that all the Xs and Ys are independent of one another. (a) Determine the distributions of X and Y. (b) Determine the distribution of X1/Sy. (c) Compute P(S< 2.393). (d) Find an interval (a, b) such that Pla < s <b) = 0.95. Make the interval such that s has equal probability of being below a as of being above b. (e) Determine the...
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...
Suppose X, Y are independent with X ∼ N (0, 1) and Y ∼ N (0, 1). Show that the distribution of Q = X/Y follows the Cauchy distribution, i.e., f(q) = 1/π(1+q2) . Hint: Let Q = X/Y and V=Y. Find the joint pdf of Q and V and finally find the marginal pdf of Q by integrating the joint pdf of Q and V w.r.t. V: Y π(1+q2) Y V = Y . Find the joint pdf of...
Suppose that the standard normal random variables X and Y are independent. Find P(0 < X<Y). 8 O 1 4T 0 1 8л Ala
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)...
Suppose that X ~ χ2(m), Y-2(n), and X and Y are independent. Is Y-X ~ χ2 if
1. Let X,.., Xn be independent and identically distributed as N (0,9) (here 9 is the variane (a) What is the distribution of Y-1X,? (Verify it using MGF) (b) What is the distribution of Xn X? (Again verify it using MGF) (c) Assume n -25. What is the probability that an observed value of X lies inside the interval [-1.2,1.2] (d) Give a lower bound on the probability that Xn lies inside the interval1.2,1.2] using Chebyshev's inequality. Compare it with...
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y < 1) =? 4、 5, P(Y < 6) =? 7, P(X + Y < 4) =?