Question

Problem 1. Let $f(x,y) = \frac{k}{8}$, $0 \le y \le 4$, $y \le x \le y+2$ be the joint pdf of X and Y. i. Find the value of k so that $f(x, y)$ is a valid joint pdf. ii. Find the marginal probability density function of X, $f_x(x)$. iii. Compute $\mu_x$, $\mu_y$, and $Cov(X, Y)$. (Hint: for $\mu_x$, use $0 \le x \le 6$).

          Problem 1. Let $f(x,y) = \frac{k}{8}$, $0 \le y \le 4$, $y \le x \le y+2$ be the joint pdf of X and Y.
i. Find the value of k so that $f(x, y)$ is a valid joint pdf.
ii. Find the marginal probability density function of X, $f_x(x)$.
iii. Compute $\mu_x$, $\mu_y$, and $Cov(X, Y)$. (Hint: for $\mu_x$, use $0 \le x \le 6$).

Problem 1. Let f(x,y) = (k)/(8), 0 ≤ y ≤ 4, y ≤ x ≤ y+2 be the joint pdf of X and Y.
i. Find the value of k so that f(x, y) is a valid joint pdf.
ii. Find the marginal probability density function of X, fx(x).
iii. Compute , , and Cov(X, Y). (Hint: for , use 0 ≤ x ≤ 6).

Added by Ryan M.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

10/07/2023

Step 1

Given that f(x,y) = c/8, 0<=y<=4, y<=x<=y+2, the integral over the entire range should be equal to 1: ∫∫ f(x,y) dx dy = 1 ∫∫ (c/8) dx dy = 1 c/8 * ∫∫ dx dy = 1 c/8 * Area of the region = 1 Since the region is a triangle with base 2 and height 4, the area is Show more…

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Let f(x,y)=(c)/(8),0<=y<=4, y<=x<=y+2 be the joint pdf of X and Y. 1: Find the value of c so that f(x,y) is a valid joint pdf. 2: Find the marginal probability density function of X, f_(X)(x). 3: Compute mu _(x),mu _(Y), and Cov(X,Y). For the bounds of mu _(x), use 0<=x<=6 Problem 1. Let f(,y) = ,0 y 4,y y+2 be the joint pdf of X andY. i. Find the value of k so that f(x,y) is a valid joint pdf. ii. Find the marginal probability density function of X, fx(x) iii. Compute x, Y, and Cov(X, Y). (Hint: for x, use 0 x 6).