Let f(x, y)=(2 / θ^2) e^-(x+y) / θ, 0<x<y<∞, zero elsewhere, be the joint pdf of the random vari

step 1 Okay to solve this question, we have been given two variables X1 and X2 are independent because

Let f(x, y)=(2 / θ^2) e^-(x+y) / θ, 0<x<y<∞, zero elsewhere,...

Let $f(x, y)=\left(2 / \theta^{2}\right) e^{-(x+y) / \theta}, 0<x<y<\infty$, zero elsewhere, be the joint pdf of the random variables $X$ and $Y$. (a) Show that the mean and the variance of $Y$ are, respectively, $3 \theta / 2$ and $5 \theta^{2} / 4$. (b) Show that $E(Y \mid x)=x+\theta$. In accordance with the theory, the expected value of $X+\theta$ is that of $Y$, namely, $3 \theta / 2$, and the variance of $X+\theta$ is less than that of $Y$. Show that the variance of $X+\theta$ is in fact $\theta^{2} / 4$.

Let $f(x, y)=\left(2 / \theta^{2}\right) e^{-(x+y) / \theta}, 0<x<y<\infty$, zero elsewhere, be the joint pdf of the random variables $X$ and $Y$.
(a) Show that the mean and the variance of $Y$ are, respectively, $3 \theta / 2$ and $5 \theta^{2} / 4$.
(b) Show that $E(Y \mid x)=x+\theta$. In accordance with the theory, the expected value of $X+\theta$ is that of $Y$, namely, $3 \theta / 2$, and the variance of $X+\theta$ is less than that of $Y$. Show that the variance of $X+\theta$ is in fact $\theta^{2} / 4$.

Key Concepts

Joint Probability Density Function

A joint probability density function (pdf) describes the likelihood of two random variables occurring simultaneously. It provides a complete description of the probabilistic behavior of both variables over a specified domain, and it is used to compute probabilities and expectations involving both variables through integration over that domain.

Marginal Distribution

The marginal distribution is obtained from a joint pdf by integrating out one of the variables. This concept is important when one wishes to study the behavior of a single variable without the influence of the other, and it is often used as a preliminary step in computing other quantities such as expectations and variances.

Law of Iterated Expectation

The law of iterated expectation, or the tower property, states that the overall expected value can be obtained by taking the expectation of the conditional expectation. This principle is integral in linking the marginal and conditional aspects of distributions and is useful in understanding hierarchical or dependent probabilistic structures.

Transformation of Random Variables and Variance Properties

Transformations of random variables involve modifying a variable by adding or multiplying by constants, where the properties of such operations (such as linearity of expectation and the effect on variance) are key. In particular, adding a constant to a random variable shifts its mean without changing its variance, and understanding this is crucial for comparing the variances of related transformed variables.

Conditional Expectation

Conditional expectation is the expected value of one random variable given that another variable is known or fixed. It is calculated using the conditional density derived from the joint pdf, and it reflects how the knowledge of one variable influences the expected outcome of the other.

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