Let X and Y be independent exponential random variables with...

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$, where $\lambda>\mu$. Let $c>0$. (a) Show that the conditional density function of $X$, given that $X+Y=c$, is $$ f_{X \mid X+Y}(x \mid c)=\frac{(\lambda-\mu) e^{-(\lambda-\mu) x}}{1-e^{-(\lambda-\mu x)}}, \quad 0<x<c $$ (b) Use part (a) and the resuit of Exercise 14 to find $E[X \mid X+Y=c]$. (c) Find $E[Y \mid X+Y=c]$.

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$, where $\lambda>\mu$. Let $c>0$.
(a) Show that the conditional density function of $X$, given that $X+Y=c$, is
$$
f_{X \mid X+Y}(x \mid c)=\frac{(\lambda-\mu) e^{-(\lambda-\mu) x}}{1-e^{-(\lambda-\mu x)}}, \quad 0<x<c
$$
(b) Use part (a) and the resuit of Exercise 14 to find $E[X \mid X+Y=c]$.
(c) Find $E[Y \mid X+Y=c]$.

Key Concepts

Conditional Expectation

Conditional expectation is the expected value of a random variable given that a certain condition, such as a fixed sum of other variables, is met. It is calculated by integrating the product of the variable and its conditional density over the relevant range. This concept is crucial for determining quantities like the expected value of one variable conditioned on the sum of two independent random variables.

Transformation of Random Variables

When dealing with functions of random variables, such as sums, it is often useful to perform a change of variables, transforming the joint distribution into a new set of variables for which the problem is easier to solve. This technique, which includes accounting for the Jacobian of the transformation, is essential for deriving the conditional density function of one variable given a function of multiple variables.

Independence of Random Variables

Independence of random variables implies that the joint probability density function factors into the product of the marginal densities. This property significantly simplifies the analysis when dealing with functions of such variables, for instance, when computing joint or conditional distributions, because the behavior of one variable does not influence the other.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is defined by a rate parameter that governs the decay of its probability density function. A key property of the exponential distribution is its memoryless nature, meaning that the probability of an event occurring in the future is independent of how much time has already elapsed.

Conditional Density Functions

A conditional density function specifies the probability distribution of one variable given that a condition involving one or more other variables is satisfied. In this context, it involves expressing the density of one random variable given that the sum of the variables is a fixed value. The conditional density is obtained by dividing the joint density by the marginal density of the conditioning variable, effectively focusing on the distribution in the constrained subspace.

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