The random variable whose probability density function is...

The random variable whose probability density function is given by $$ f(x)=\left\{\begin{array}{ll} \frac{1}{2} \lambda e^{\lambda x}, & \text { if } x \leqslant 0 \\ \frac{1}{2} \lambda e^{-\lambda x}, & \text { if } x>0 \end{array}\right. $$ is said to have a Laplace, sometimes called a double exponential, distribution. (a) Verify that the preceding is a probability density function. (b) Show that the distribution function of a Laplace random variable is $$ F(x)=\left\{\begin{array}{ll} \frac{1}{2} e^{\lambda x}, & \text { if } x \leqslant 0 \\ 1-\frac{1}{2} e^{-\lambda x}, & \text { if } x>0 \end{array}\right. $$ Let $X$ and $Y$ be independent exponential random variables with parameter $\lambda$. Also, let $I$ be independent of $X$ and $Y$ and let it be equally likely to be 1 or $-1$. (c) Show that $X-Y$ is a Laplace random variable. (d) Show that $I X$ is a Laplace random variable. (e) Show that $W$ is a Laplace random variable, where $$ W=\left\{\begin{array}{ll} X, & \text { if } I=1 \\ -Y, & \text { if } I=-1 \end{array}\right. $$

The random variable whose probability density function is given by
$$
f(x)=\left\{\begin{array}{ll}
\frac{1}{2} \lambda e^{\lambda x}, & \text { if } x \leqslant 0 \\
\frac{1}{2} \lambda e^{-\lambda x}, & \text { if } x>0
\end{array}\right.
$$
is said to have a Laplace, sometimes called a double exponential, distribution.
(a) Verify that the preceding is a probability density function.
(b) Show that the distribution function of a Laplace random variable is
$$
F(x)=\left\{\begin{array}{ll}
\frac{1}{2} e^{\lambda x}, & \text { if } x \leqslant 0 \\
1-\frac{1}{2} e^{-\lambda x}, & \text { if } x>0
\end{array}\right.
$$
Let $X$ and $Y$ be independent exponential random variables with parameter $\lambda$. Also, let $I$ be independent of $X$ and $Y$ and let it be equally likely to be 1 or $-1$.
(c) Show that $X-Y$ is a Laplace random variable.
(d) Show that $I X$ is a Laplace random variable.
(e) Show that $W$ is a Laplace random variable, where
$$
W=\left\{\begin{array}{ll}
X, & \text { if } I=1 \\
-Y, & \text { if } I=-1
\end{array}\right.
$$

Key Concepts

Transformation of Random Variables

Transformation of random variables involves deriving the distribution of a new variable defined as a function of one or more random variables. This concept is crucial when operations such as addition, subtraction, or sign changes (e.g., multiplying by a random sign indicator) are applied, leading to distributions that might have different shapes or properties from the original variables.

Laplace (Double Exponential) Distribution

The Laplace distribution, also known as the double exponential distribution, is characterized by a probability density that is symmetric around a central point with exponential decay on both sides. Its key properties include a sharp peak at the mean and heavier tails compared to the normal distribution, and it arises in various contexts, especially in scenarios involving differences of exponential variables or sign-flipped variables.

Independence of Random Variables

Independence in probability means that the occurrence of one event or the value of one random variable does not affect the probability distribution of another. This property is important when working with joint distributions, as it allows for the factorization of joint densities and simplifies the analysis of transformed random variables, such as when combining exponential and indicator variables.

Probability Density Function (PDF)

A probability density function is a function that describes the relative likelihood for a continuous random variable to take on a given value. A key property of any PDF is that its integral over the entire space equals 1, ensuring that it represents a valid probability model.

Cumulative Distribution Function (CDF)

The cumulative distribution function of a random variable provides the probability that the variable will take a value less than or equal to a given point. It is derived by integrating the PDF and is essential for understanding the distribution’s behavior and for computing probabilities over intervals.

Please give Ace some feedback