Let \(f(x, y)\) be a joint probability density function, i.e., \(f(x, y) d x d y\) is the probability that \(X\) lies between \(x\) and \(x+d x\) and \(Y\) lies between \(y\) and \(y+d y\). If \(X\) and \(Y\) are independent, then $$ f(x, y) d x d y=f_{1}(x) f_{2}(y) d x d y . $$ Therefore, if \(W=X+Y\), show that $$ \bar{W}=\bar{X}+\bar{Y} $$ and that $$ \overline{(W-\bar{W})^{2}}=\overline{(X-\bar{X})^{2}}+\overline{(Y-\bar{Y})^{2}} . $$ In other words, if \(X\) and \(Y\) are independent, the mean and variance of their sum is equal to the sum of their means and variances.

The expected value of a continuous random variable is a fundamental concept in probability that represents the average value the variable is predicted to attain over a large number of observations. Mathematically, for a random variable with a probability density function (pdf) \(f(x)\), the expected value, also known as the expectation or mean, is calculated using an integral over the entire range of possible values.

Intuitively, you can think of the expected value as the 'balance point' of the distribution of the random variable. To calculate it, you multiply each possible value \(x\) of the random variable by its probability density \(f(x)\), then sum all these products across the range of \(x\). This is analogous to calculating the center of mass in a physical distribution.

In terms of calculation for a single random variable \(X\), the expected value \(\bar{X}\) is given by the integral:\r\[\bar{X} = \int_{-\infty}^{\infty} x f(x) dx.\]
When dealing with joint probability densities of two independent variables, like \(X\) and \(Y\), you can find the mean of their sum \(W=X+Y\) by adding their individual means: \(\bar{W} = \bar{X} + \bar{Y}\). This is because the expected value of the sum of independent variables is equal to the sum of their expected values.

The mean, or expected value, of a random variable is a cornerstone in understanding its behavior. It provides a single summarizing number for the central location of that variable's distribution. Specifically, the mean of a continuous random variable captures where most values are expected to lie if we were to continuously sample from that variable.

To calculate the mean \(\bar{X}\), you integrate the product of the value and its likelihood according to the probability density function across all possible values:\r\[\bar{X} = \int_{-\infty}^{\infty} x f(x) dx.\]
This process is akin to finding the average, but in the context of probability distributions, it accounts for the continuous nature of the data and the varying probabilities of different outcomes.

Variance is a quantitative measure capturing the spread or variability of a random variable's distribution. Unlike the mean, which focuses on the central tendency, variance looks at how far the values are dispersed from the mean.

The formal calculation for the variance of a continuous random variable \(X\) with mean \(\bar{X}\) and probability density function \(f(x)\) is:\r\[\overline{(X-\bar{X})^2} = \int_{-\infty}^{\infty} (x-\bar{X})^2 f(x) dx.\]
In practice, this means you are averaging the squared differences between the random variable's values and its mean. High variance indicates that the numbers are more spread out, while low variance suggests they are closer to the mean. For a sum of independent random variables (like \(W = X + Y\)), the variance is additive: \(\overline{(W-\bar{W})^2} = \overline{(X-\bar{X})^2} + \overline{(Y-\bar{Y})^2}\). This highlights that independent variables' variabilities do not influence each other.

Random variables are independent if the occurrence of one does not affect the probability of occurrence of the other. This property greatly simplifies many operations in probability. For instance, when random variables \(X\) and \(Y\) are independent, their joint probability density function can be expressed as the product of their individual densities, \(f(x, y) = f_1(x)f_2(y)\).

This independence has important implications for statistical measures like the mean and variance. As seen in the exercise, the mean and variance of the sum of independent random variables are simply the sums of their respective means and variances:

• The mean of the sum: \(\bar{W} = \bar{X} + \bar{Y}\)
• The variance of the sum: \(\overline{(W-\bar{W})^2} = \overline{(X-\bar{X})^2} + \overline{(Y-\bar{Y})^2}\)

Understanding independence is crucial for analyzing complex systems by considering their components separately.