Question

Consider the following joint probability density function of the random variables $X$ and $Y$ : $$ f(x, y)=\left\{\begin{array}{ll} \frac{3 x-y}{11}, & 1<x<3,0<y<1, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Find the marginal density functions of $X$ and $Y$. (b) Are $X$ and $Y$ independent? (c) Find $P(X>2)$

          Consider the following joint probability density function of the random variables $X$ and $Y$ :
$$
f(x, y)=\left\{\begin{array}{ll}
\frac{3 x-y}{11}, & 1<x<3,0<y<1, \\
0, & \text { elsewhere. }
\end{array}\right.
$$
(a) Find the marginal density functions of $X$ and $Y$.
(b) Are $X$ and $Y$ independent?
(c) Find $P(X>2)$

Added by Shawn Z.

Probability with Applications in Engineering, Science, and Technology

Matthew A. Carlton • Jay L. Devore 2nd Edition

Instant Answer

Solved by Expert Adi S

09/01/2022

Step 1

Given $f(x, y) = \frac{3x - y}{11}$ for $1 < x < 3$ and $0 < y < 1$, we need to find $f_X(x) = \int_{0}^{1} f(x, y) dy$. $f_X(x) = \int_{0}^{1} \frac{3x - y}{11} dy = \frac{3x}{11}y - \frac{y^2}{2 \cdot 11} \Big|_{0}^{1} = \frac{3x}{11} - \frac{1}{22}$ for $1 < x Show more…

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Consider the following joint probability density function of the random variables $X$ and $Y$ : $$ f(x, y)=\left\{\begin{array}{ll} \frac{3 x-y}{11}, & 1<x<3,0<y<1, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Find the marginal density functions of $X$ and $Y$. (b) Are $X$ and $Y$ independent? (c) Find $P(X>2)$