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 $P(X>2)$

Key Concepts

Independence of Random Variables

Two random variables are said to be independent if the joint PDF factors into the product of the marginal PDFs. This means that the occurrence of one variable's event does not provide any information about the other. Checking this property involves comparing the joint PDF to the product of the marginals over the domain of interest.

Probability Computation for Continuous Variables

Computing probabilities for continuous random variables typically involves integrating the density function over the region corresponding to the event. This method is used to determine the likelihood of events defined by inequalities, such as calculating the probability that a variable exceeds a certain value.

Joint Probability Density Function

A joint probability density function (PDF) describes the likelihood of two continuous random variables taking on specific values simultaneously. It must be non-negative everywhere and its integral over the entire space must equal 1. This concept is fundamental for understanding how two variables behave together and is the starting point for deriving other properties such as marginal distributions and independence.

Marginal Density Function

The marginal density function of a random variable is obtained by integrating the joint PDF over the range of the other variable. It provides the probabilities associated with one variable irrespective of the values of the other variable. This process, known as marginalization, is essential in reducing a joint distribution to a single-variable distribution.

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