ISI MStat PSB 2012 Problem 3 | Finding the Distribution of a Random Variable

This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 3 based on finding the distribution of a random variable . Let's give it a try !!

Problem- ISI MStat PSB 2012 Problem 3

Let $X_{1}$ and $X_{2}$ be i.i.d. exponential random variables with mean $\lambda>0$ .Let $Y_{1}=X_{1}-X_{2}$ and $Y_{2}=R X_{1}-(1-R) X_{2},$ where $R$ is a Bernoulli random variable with parameter $1 / 2$ and is independent of $X_{1}$ and $X_{2}$
(a) Show that $Y_{1}$ and $Y_{2}$ have the same distribution.
(b) Obtain the common density function.

Prerequisites

Cumulative Distribution Function

Bernoulli distribution

Exponential Distribution

Solution :

Cumulative distribution of $Y_{1}$ be

$F_{Y_{1}}(y_{1})=P(Y_{1} \leq y_{1})=P(x_{1}-x_{2} \leq y_{1})$ , $y_1 \in R$

$=P(x_{1} \leq y_{1}+x_{2})$
Now, $y_{1}+x_{2} \ge 0 \Rightarrow x_{2} \ge-y_{1}$
Now, if $y_{1} \ge 0$ then,
$P(x_{1} \le y_{1}+x_{2}) =\int_{0}^{\infty} P(x_{1} \le y_{1}+x_{2}t) \lambda e^{-\lambda x_{2}} d x_{2}$

= $\int_{0}^{\infty} \int_{0}^{y_{1}+x_{2}} \lambda e^{-\lambda x_{1}} x \lambda e^{-\lambda x_{2}} d x_{1} d x_{2}$

= $\int_{0}^{\infty} \lambda e^{\lambda x_{2}} x \lambda \times \frac{1}{\lambda} (1-e^{-(\lambda y_{1}+x_{2}) }) d x_{2}$

= $\int_{0}^{\infty} \lambda e^{-\lambda x_{2}} d x_{2}-\int_{0}^{\infty} \lambda e^{-\lambda (y_{1}+2 x_{2})} d x_{2}$

= $1-\frac{e^{-\lambda y_{1}}}{2}$

Now, $y_{1} \le 0$ then,
$P(x_{1} \leq y_{1}+x_{2}) =\int_{-y_{1}}^{\infty} \int_{0}^{y_{1}+x_{2}} \lambda e^{-\lambda x_{4}} x \lambda e^{-\lambda x_{2}} d x_{1} d x_{2}$
$=\int_{-y_{1}}^{\infty} x e^{-\lambda x_{2}}(1-e^{-\lambda(y_{1}+x_{1})}) d x_{2}$
$=\lambda \int_{-y_{1}}^{\infty} e^{-x^{2} x_{2}} d x_{2}-\int_{-y_{1}}^{\infty} \lambda e^{-\lambda(y_{1}+2 x_{2})} d x_{2}$
$=e^{+\lambda y_{1}}-\frac{e^{-\lambda y_{1}}}{2} x e^{+2 \lambda y_{1}}$
$=\frac{e^{\lambda y_{1}}}{2}$
Therefore, $F_{Y_{1}}(y_{1}) = \begin{cases} 1-\frac{e^{-\lambda y_{1}}}{2} & , i f y_{1} \ge 0 \\ \frac{e^{\lambda y_{1}}}{2} & ,if y_{1}<0 \end{cases}.$

Cumulative distribution of $Y_{2}$ be $F_{Y_{2}}(y_{2})=P(Y_{2} \le y_{2})$ , $y_2 \in R$

= $P(Y_{2} \le y_{2} \mid R=1) P(R=1)+P(Y_{2} \le y_{2} \mid R=0) P(R=0)$
$=P(x_{1} \le y_{2}) \times \frac{1}{2}+P(-x_{2} \le y_{2}) \times \frac{1}{2}$
= $\begin{cases} \frac{1}{2} [F_{x_{1}}(y_{2})+1] & , y_{2} \ge 0 \\ \frac{1}{2} [1-F_{x_{2}}(-y_{2})] & ,y_{2}<0 \end{cases}.$
= $\begin{cases} 1-\frac{e^{-\lambda y_{2}}}{2}, & \text { if } y_{2} \ge 0 \\ \frac{e^{\lambda y_{2}}}{2} \end{cases}.$
since cdf of exponential random Variable, X is $(1-e^{-\lambda x}), x \ge 0$
Thus both $Y_{1}$ and $Y_{2}$ has same distribution
(b) $f_{Y_{1}}(y_{1})=\begin{cases} \frac{d}{d y_{1}}(1-\frac{e^{-\lambda y_{1}}}{2}) & \text { if } y_{1} \ge 0 \\ \frac{d}{d y_{1}}(\frac{e^{\lambda y_{1}}}{2}) & , \text { if } y_{2}<0 \end{cases}$

= $\begin{cases} \frac{\lambda e^{-\lambda y_{1}}}{2} & \text { if } y_{1} \ge 0 \\ \frac{\lambda e^{\lambda y_{1}}}{2} & , \text { if } y_{1}<0 \end{cases}$

Similarly, for $Y_2$ .

Food For Thought

If $\theta \sim U(0, 2 \pi )$ then find the distribution of $sin(\theta + {\theta}_{0} )$ , where ${\theta}_{0} \in (0,2 \pi)$ .

Outstanding Statistics Program with Applications