Question

Suppose X and Y have a joint pdf f given by f(x, y) = { ce^{-y+x^2} -1/2 ? x ? 1/2, y > x^2, 0 otherwise. (a) Determine the value of c. (b) Calculate the marginal probability density function of X. (c) Only use cumulative distribution functions in your answer to the following question: are X and Y independent? (d) Calculate the conditional probability density function of Y given X = x, for -1/2 ? x ? 1/2. (e) Calculate E(Y|X = x) for -1/2 ? x ? 1/2.

          Suppose X and Y have a joint pdf f given by
f(x, y) = {
ce^{-y+x^2}  -1/2 ? x ? 1/2, y > x^2,
0  otherwise.
(a) Determine the value of c.
(b) Calculate the marginal probability density function of X.
(c) Only use cumulative distribution functions in your answer to the following question: are X and Y independent?
(d) Calculate the conditional probability density function of Y given X = x, for -1/2 ? x ? 1/2.
(e) Calculate E(Y|X = x) for -1/2 ? x ? 1/2.

Suppose X and Y have a joint pdf f given by
f(x, y) =
ce^-y+x^2 -1/2 ? x ? 1/2, y > x^2,
0 otherwise.
(a) Determine the value of c.
(b) Calculate the marginal probability density function of X.
(c) Only use cumulative distribution functions in your answer to the following question: are X and Y independent?
(d) Calculate the conditional probability density function of Y given X = x, for -1/2 ? x ? 1/2.
(e) Calculate E(Y|X = x) for -1/2 ? x ? 1/2.

Added by -Ngeles B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

02/24/2023

Step 1

So, we have: ∫∫f(x, y) dy dx = c ∫∫(y + z) dy dx = 1 The limits of integration for x are given as -1/2 < x < 1/2. The limits of integration for y are given as y > x^2, which means y goes from x^2 to infinity. So, we have: c ∫[-1/2, 1/2] ∫[x^2, ∞] (y + z) dy dx Show more…

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Suppose X and Y have a joint pdf f given by f(x,y) = ce^{-y+x^2} for -1/2 ≤ x ≤ 1/2, y > x^2, 0 otherwise. (a) Determine the value of c. (b) Calculate the marginal probability density function of X. (c) Only use cumulative distribution functions in your answer to the following question: are X and Y independent? (d) Calculate the conditional probability density function of Y given X = x, for -1/2 ≤ x ≤ 1/2. (e) Calculate E(Y|X = x) for -1/2 ≤ x ≤ 1/2.