Consider the random variables X and Y with joint density...

Consider the random variables $X$ and $Y$ with joint density function
$$
f(x, y)=\left\{\begin{array}{ll}
x+y, & 0 \leq x, y \leq 1, \\
0, & \text { elsewhere. }
\end{array}\right.
$$
(a) Find the marginal distributions of $X$ and $Y$.
(b) Find $P(X>0.5, Y>0.5)$

Key Concepts

Joint Probability Density Function

A joint probability density function (PDF) describes the likelihood of two random variables taking on a set of values simultaneously over their range. It is fundamental in multivariate probability theory and must satisfy the properties of being non-negative and integrating to one over its entire support.

Marginal Distribution

The marginal distribution of a random variable is obtained by integrating the joint PDF over the range of the other variable. This operation 'marginalizes out' the other variable, yielding the probability distribution of the variable of interest, which is essential when one is looking at single-variable behaviors in a multivariate setting.

Probability Calculation over a Region

Computing the probability of an event defined by a region in the variables' domain involves setting up a double integral of the joint PDF over that specific region. This concept is crucial for determining the likelihood of simultaneous events, such as both variables exceeding certain thresholds.

Double Integration in Bounded Regions

When the joint PDF is defined over a bounded region, double integration techniques are used. This means integrating first with respect to one variable while treating the other as constant, and then integrating the resulting function with the other variable. This method is key in evaluating marginal distributions and event probabilities for continuous random variables.

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