Find the covariance of random variables X and Y having the...

Find the covariance of random variables $X$ and $Y$ having the joint probability density function
$$
f(x, y)=\left\{\begin{array}{ll}
x+y, & 0<x<1,0<y<1 \\
0, & \text { elsewhere }
\end{array}\right.
$$

Key Concepts

Expected Value

The expected value of a random variable is the long-run average value of repetitions of the experiment it represents. For continuous variables, it is calculated as the integral of the variable multiplied by its probability density function over its entire range. In the context of joint distributions, expectations such as E[XY] are found by integrating the product of the variables and the joint pdf over their joint support.

Marginal Distribution

The marginal distribution of a random variable is obtained by integrating the joint probability density function over the range of the other variable. This process provides the probability distribution of one variable irrespective of the other, and it is used to calculate quantities such as E[X] and E[Y] individually.

Covariance

Covariance is a measure of the joint variability of two random variables. It quantifies the degree to which the two variables vary together, and is defined as the expected value of the product of their deviations from their respective means. Specifically, the covariance is computed using the formula Cov(X, Y) = E[XY] – E[X]E[Y].

Joint Probability Density Function

A joint probability density function (pdf) describes the probability distribution of two continuous random variables simultaneously. It assigns probabilities to pairs of outcomes, and its integral over any region of the variables’ support yields the probability that the pair falls within that region.

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