Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y...

Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y + 1, be the joint pdf of X and Y.

(a) (3 pts) Find c and sketch the region for which f (x, y) > 0.

(b) (3 pts) Find f_X(x), the marginal pdf of X.

(c) (3 pts) Find f_Y(y), the marginal pdf of Y.

(d) (3 pts) Find P(X ≤ 3 − Y).

(e) (4 pts) E(X) and Var(X).

(f) (4 pts) E(Y) and Var(Y).

(g) (3 pts) Cov(X,Y).

(h) ( 3 pts) Find ρ, the correlation coefficient of X and Y.

(i) (3 pts) Are X and Y independent or dependent? Why or why not?

(j) (3 pts) Determine h(y | x), the conditional pdf of Y, given that X = x.

(k) (3 pts) Determine g(x | y), the conditional pdf of X, given that Y = y.

(l) (3 pts) Compute E(Y | x), the conditional mean of Y, given that X = x.

(m) (3 pts) Compute E(X | y), the conditional mean of X, given that Y = y.

Expert Solution

orchestra answered 1 year ago

Let X and Y have joint PDF f(x) = c(e^-(x/λ + y/μ)) 0 < x <...

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1 Let f(x, y) = 4xy, 0 < x < 1, 0 < y < 1,...

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Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2,...

Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2, f(1, 0) = 3, f(1, 1) = 5, f(2, 0) = 5, f(2, 1) = 10. Determine the Lagrange interpolation F(x, y) that interpolates the above data. Use Lagrangian bi-variate interpolation to solve this and also show the working steps.

G1. Let f(x, y) = 1 for 0 < x < 1 and x < y...

G1. Let f(x, y) = 1 for 0 < x < 1 and x < y < (x + 1); and 0 otherwise. Find the correlation coefficient for this X and Y . (Hint: the answer is p (1/2) = 0.7071. See if you know all of the steps needed to get there.)

Let X and Y have the joint pdf f(x, y) = 8xy, 0 ≤ x ≤...

Let X and Y have the joint pdf f(x, y) = 8xy, 0 ≤ x ≤ y ≤ 1. (i) Find the conditional means of X given Y, and Y given X. (ii) Find the conditional variance of X given Y. (iii) Find the correlation coefficient between X and Y.

Let the joint pmf of X and Y be defined by f (x, y) = c(x...

Let the joint pmf of X and Y be defined by f (x, y) = c(x + y), x =0, 1, 2, y = 0, 1, with y ≤ x. 1. Are X and Y independent or dependent? Why or why not? 2. Find g(x | y) and draw a figure depicting the conditional pmfs for y =0 and 1. 3. Find h(y | x) and draw a figure depicting the conditional pmfs for x = 0, 1 and2. 4....

Let the joint pmf of X and Y be defined by f (x, y) = c(x...

Let the joint pmf of X and Y be defined by f (x, y) = c(x + y), x =0, 1, 2, y = 0, 1, with y ≤ x. 1. Find g(x | y) and draw a figure depicting the conditional pmfs for y =0 and 1. 2. Find h(y | x) and draw a figure depicting the conditional pmfs for x = 0, 1 and2. 3. Find P(0 < X <2 |Y = 0), P(X ≤ 2 |Y...

The joint PDF of X and Y is given by f(x, y) = C, (0< x<y<1)....

The joint PDF of X and Y is given by f(x, y) = C, (0< x<y<1). a) Determine the value of C b) Determine the marginal distribution of X and compute E(X) and Var(X) c) Determine the marginal distribution of Y and compute E(Y) and Var(Y) d) Compute the correlation coefficient between X and Y

Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and 0<=y<=1. a) Find k. b)...

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Question

Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y...

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