Use convolution to show that if X and Y are independent...

Use convolution to show that if $X$ and $Y$ are independent exponential random variables with parameters $\alpha$ and $\beta$, respectively, where $\alpha \neq \beta$, then the density function $f$ of their sum $X+Y$ is given by $$ f_{X+Y}(t)=\frac{\alpha \beta}{\alpha-\beta}\left(e^{-\beta t}-e^{-\alpha t}\right), \quad t \geq 0 . $$ [We showed this, using the Laplace-Stieltjes transform, in Example 3.4.1.]

Use convolution to show that if $X$ and $Y$ are independent exponential random variables with parameters $\alpha$ and $\beta$, respectively, where $\alpha \neq \beta$, then the density function $f$ of their sum $X+Y$ is given by
$$
f_{X+Y}(t)=\frac{\alpha \beta}{\alpha-\beta}\left(e^{-\beta t}-e^{-\alpha t}\right), \quad t \geq 0 .
$$
[We showed this, using the Laplace-Stieltjes transform, in Example 3.4.1.]

Key Concepts

Probability Density Function

A probability density function (pdf) describes the likelihood of a continuous random variable taking a specific value. In the context of summing independent random variables, the pdf of the sum is found by convolving the individual pdfs, which involves integrating the product of these functions over the field where they overlap.

Convolution

Convolution is an integral transform that combines two functions to produce a third function. In probability theory, the convolution of two independent random variables' density functions yields the density function of their sum. This technique, executed via an integral over an appropriate domain, is essential for deriving the combined effect of two independent distributions.

Exponential Distribution

The exponential distribution is a continuous probability distribution characterized by its rate parameter. It is notable for its memoryless property and is often used to model the time between independent events. Understanding its probability density function is crucial when dealing with sums of exponential random variables.

Independence

Independence is a fundamental concept in probability meaning that the outcome of one random variable does not affect the outcome of another. This property allows the joint density of independent random variables to be expressed as the product of their individual densities, a key step in calculating the convolution.

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