Consider the following joint probability density function of...

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 $P(X>2)$

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Key Concepts

Joint Probability Density Function (pdf)

A joint probability density function describes the likelihood of two continuous random variables taking on a particular pair of values. It integrates to 1 over its support and defines the behavior of the variables, including any inherent relationships or dependencies they might have.

Marginal Density Functions

Marginal density functions are derived by integrating the joint density over the range of the other variable. This process isolates the probability distribution of a single variable without considering the joint behavior, thereby simplifying the analysis of individual random variables.

Independence of Random Variables

Random variables are independent if the joint density function can be expressed as the product of their marginal density functions. This property is fundamental in probability theory as it implies that the realization of one variable does not affect the likelihood of outcomes of the other.

Probability Calculation using Marginal Densities

Calculating the probability of an event involving one of the random variables involves integrating the marginal density function over the specific range corresponding to the event. This technique simplifies probability computations when dealing with one-dimensional distributions derived from the joint pdf.

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