Question

Let \(f(x, y)=\left(2 / \theta^{2}\right) e^{-(x+y) / \theta}, 0

Short answer

The mean and variance of \(Y\) are \(3\theta/2\) and \(5\theta^2/4\) respectively. The expectation of \(Y\) given \(X=x\) is \(x+\theta\). The variance of \(X+\theta\) is \(\theta^2/4\).

Step-by-step solution

Calculating the Mean of Y

The mean or expected value of \(Y\) can be calculated via the integral \(\int_{-\infty}^{+\infty} y\cdot f(y) dy\), where \(f(y)\) is the marginal pdf of \(Y\). As \(f(x, y) = (2 / \theta^{2}) e^{-(x+y) / \theta}, 0<x<y<\infty\), we first need to get the marginal pdf \(f(y) = \int_{0}^{y}(2 / \theta^{2}) e^{-(x+y) / \theta} dx\). After getting \(f(y)\), we substitute it into the formula for expectation of \(Y\), and solve the integral.

Calculating the Variance of Y

Variance of a random variable, \(Var(Y) = E(Y^2) - (E(Y))^2\). Thus, we first need to calculate \(E(Y^2)\), which can be computed from \(\int_{-\infty}^{+\infty} y^{2}\cdot f(y) dy\), by substituting marginal pdf \(f(y)\) computed in step 1. Then calculate \((E(y))^2\). The difference between the two values gives the variance.

Calculating the Conditional Expectation E(Y|X=x)

The conditional expectation \(E(Y|X=x)\) can be calculated by integrating \(y*f(y|x)dy\) over all possible \(y\), where \(f(y|x)\) is the conditional pdf of \(Y\) given \(X=x\). But in our case, we already know from the theory that \(E(Y|X=x) = x + \theta\).

Calculating the Variance of X + theta

The formula for variance is \(Var(aX+b) = a^2 Var(X)\) for constants \(a\) and \(b\). By letting \(a=1\) and \(b=\theta\), we have \(Var(X+\theta) = Var(X)\), and thus only need to calculate the variance of \(X\). Once that is obtained, the comparison to the variance of \(Y\) can be conducted.

Key concepts

Expected Value

The expected value, often represented as E(X), is a fundamental concept in probability, referring to the long-run average value of repetitions of the experiment it represents. It is calculated as the sum of all possible values of the random variable, each multiplied by its probability of occurrence, or through the integral of the variable's probability density function over its range. In the case of a continuous random variable, such as in our exercise, the expected value of Y would be calculated by integrating the product of y and its marginal probability density function (pdf), represented as \(E(Y) = \int_{-\infty}^{+\infty} y \cdot f(y) dy\). Once computed, the expected value provides a measure of the central tendency or the 'center' of the distribution of the random variable.
When dealing with joint probability density functions, to find the expected value of a single variable, we first need to derive its marginal pdf by integrating out the other variable from the joint pdf, a process clearly outlined in the example exercise.

Variance of a Random Variable

Variance, denoted as Var(X), is a measure that tells us how much the values of a random variable X are spread out from their expected value. It is computed as the expected value of the squared deviation from the mean, mathematically given by \(Var(X) = E[(X - E(X))^2]\). This can also be simplified to \(Var(X) = E(X^2) - [E(X)]^2\), the formula used in our exercise.
The variance is a crucial concept in statistics because it quantifies the extent to which a set of numbers is dispersed around the mean. A larger variance implies more spread out data. In the present exercise, after computing the expected value of Y (\(E(Y)\)), the next step involves calculating \(E(Y^2)\) to eventually find Var(Y). These calculations shed light on the uncertainty or ‘risk’ associated with the random variable Y.

Conditional Expectation

Conditional expectation, expressed as \(E(Y | X = x)\), is the expected value of a random variable Y given that another random variable X has a certain value. This concept is critical when we have joint distributions, as it provides a way to understand one variable in the context of another. In terms of calculation, for a continuous random variable, it involves integrating the product of y and the conditional pdf of Y given X equals x over all y values, which can be complex.
In our textbook exercise, the conditional expectation \(E(Y | X)\) simplifies to a more straightforward expression \(x + \theta\), demonstrating a linear relationship between Y and X modified by \(\theta\). Understanding how these variables interact is key to mastering conditional probability and expectation, which often has practical implications in fields such as finance and engineering.

Marginal Probability Density Function

The marginal probability density function (pdf) provides the probabilities of a single random variable regardless of the values of other variables. For continuous random variables in a joint pdf, like in our example, the marginal pdf of Y, denoted as \(f(y)\), is found by integrating the joint pdf over the range of X, effectively 'summing out' X's influence.
Obtaining the marginal pdf is often the first step in finding other properties such as expected value or variance of that variable. As shown in the step-by-step solution, calculating \(f(y)\) is essential before we can proceed with computing E(Y) and Var(Y). The marginal pdf reflects the individual behavior of the variable, giving us insight into its standalone probability structure without considering its dependence on other variables.