Suppose the random variables X and Y form a bivariate normal distribution. You are given that...

Suppose the random variables X and Y form a bivariate normal distribution. You are given that E[X] = 3, E[Y ] = −2, σX = 4, and σY = 3. Find the probability that X and Y are within 3 of each other under the following additional assumptions:

(a) Corr(X, Y ) = 0

(b) Corr(X, Y ) = −0.6

Expert Solution

orchestra answered 1 year ago

Given below is a bivariate distribution for the random variables x and y. f(x, y) x...

Given below is a bivariate distribution for the random variables x and y. f(x, y) x y 0.3 50 80 0.2 30 50 0.5 40 60 (a) Compute the expected value and the variance for x and y. E(x) = E(y) = Var(x) = Var(y) = (b) Develop a probability distribution for x + y. x + y f(x + y) 130 80 100 (c) Using the result of part (b), compute E(x + y) and Var(x + y). E(x...

Suppose the joint probability distribution of two binary random variables X and Y are given as...

Suppose the joint probability distribution of two binary random variables X and Y are given as follows. x/y 1 2 0 3/10 0 1 4/10 3/10 X goes along side as 0 and 1, Y goes along top as 1 and 2 e) Find joint entropy H(X, Y ). f) Suppose X and Y are independent. Show that H(X|Y ) = H(X). g) Suppose X and Y are independent. Show that H(X, Y ) = H(X) + H(Y ). h)...

Suppose that X and Y are standard bivariate normal with correlation ρ = 0.4. You will...

Suppose that X and Y are standard bivariate normal with correlation ρ = 0.4. You will need to use the above table and compute each of the following: (a) P(X ≥ 1). (b) P(X ≥ 1 | Y = 1). (c) P(X ≥ 1 | Y = −1). x 0.6547 1 1.5275 Φ(x) 0.7437 0.8413 0.9367

X and Y are discrete random variables with joint distribution given below Y 1 Y 0...

X and Y are discrete random variables with joint distribution given below Y 1 Y 0 Y 1 X 1 0 1/4 0 X 0 1/4 1/4 1/4 (a) Determine the conditional expectation E Y|X 1 . (b) Determine the conditional expectation E X|Y y for each value of y. (c) Determine the expected value of X using conditional expectation results form part (b) above. (d) Now obatin the marginal distribution of X and verify your answers to part (c).

A: Suppose two random variables X and Y are independent and identically distributed as standard normal....

A: Suppose two random variables X and Y are independent and identically distributed as standard normal. Specify the joint probability density function f(x, y) of X and Y. Next, suppose two random variables X and Y are independent and identically distributed as Bernoulli with parameter 1 2 . Specify the joint probability mass function f(x, y) of X and Y. B: Consider a time series realization X = [10, 15, 23, 20, 19] with a length of five-periods. Compute the...

This is the probability distribution between two random variables X and Y: Y \ X 0...

This is the probability distribution between two random variables X and Y: Y \ X 0 1 2 3 0.1 0.2 0.2 4 0.2 0.2 0.1 a) Are those variables independent? b) What is the marginal probability of X? c) Find E[XY]

Suppose X and Y are independent random variables with Exp(θ = 2) distribution. Note that, we...

Suppose X and Y are independent random variables with Exp(θ = 2) distribution. Note that, we say X ∼ Exp(θ) if its pdf is f(x) = 1/θ e^(−x/θ) , for x > 0 and θ > 0. (a) What is the joint probability density function (pdf) of (X, Y )? (b) Use the change of variable technique (transformation technique) to evaluate the joint pdf fW,Z (w, z) of (W, Z), where W = X −Y and Z = Y ....

Let X and Y be jointly normal random variables with parameters E(X) = E(Y ) =...

Let X and Y be jointly normal random variables with parameters E(X) = E(Y ) = 0, Var(X) = Var(Y ) = 1, and Cor(X, Y ) = ρ. For some nonnegative constants a ≥ 0 and b ≥ 0 with a2 + b2 > 0 define W1 = aX + bY and W2 = bX + aY . (a)Show that Cor(W1, W2) ≥ ρ (b)Assume that ρ = 0.1. Are W1 and W2 independent or not? Why? (c)Assume now...

Let X and Y be two discrete random variables whose joint probability distribution function is given...

Let X and Y be two discrete random variables whose joint probability distribution function is given by f(x, y) = 1 12 x 2 + y for x = −1, 1 and y = 0, 1, 2 a) Find cov(X, Y ). b) Determine whether X and Y are independent or not. (Justify your answer.

Given X + Y has a normal distribution, prove that X and Y each have a...

Given X + Y has a normal distribution, prove that X and Y each have a normal distribution if X and Y are iid random variables

Question