Practice Problems in Random Variables

Problem Set 1

Conditional Distribution of a Random Variable

It describes how the distribution of a random variable changes when we are given additional information (i.e., when an event has occurred).

Problem Set 2

Functions of One Random Variable

Distribution of $Y=g(X)$

Monotonic transformations (either increasing or decreasing.. going in one direction)

Non-monotonic transformations

Distribution of $Y=X^2$

Method using CDF

Method using PDF (Jacobian technique)

Piecewise transformations

🔹 Problem 2 – Non-Monotonic Transformation

Let $X\sim \text{Uniform}(-1,1)$

Find the PDF of $Y=X^2.$

To find the probability density function (PDF) of $Y = X^2$ where $X \sim \text{Uniform}(-1, 1)$ , we can use the method of transformations (the Cumulative Distribution Function method).

1. Identify the distribution of $X$

The PDF of $X$ is given by:

$f_X(x) = \begin{cases} \frac{1}{2} & \text{if } -1 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$

2. Determine the range of $Y$

Since $X$ takes values in the interval $[-1, 1]$ , the variable $Y = X^2$ takes values in the interval $[0, 1]$ .

3. Cumulative Distribution Function (CDF) Method

For $y \in [0, 1]$ , the CDF of $Y$ is:

$F_Y(y) = P(Y \leq y) = P(X^2 \leq y)$

Since $y \geq 0$ , $X^2 \leq y$ is equivalent to $-\sqrt{y} \leq X \leq \sqrt{y}$ . Thus:

$F_Y(y) = P(-\sqrt{y} \leq X \leq \sqrt{y})$

Integrating the PDF of $X$ over this interval:

$F_Y(y) = \int_{-\sqrt{y}}^{\sqrt{y}} f_X(x) \, dx = \int_{-\sqrt{y}}^{\sqrt{y}} \frac{1}{2} \, dx$

$F_Y(y) = \frac{1}{2} \left[ x \right]_{-\sqrt{y}}^{\sqrt{y}} = \frac{1}{2} (\sqrt{y} - (-\sqrt{y})) = \frac{1}{2} (2\sqrt{y}) = \sqrt{y}$

4. Find the PDF of $Y$

The PDF $f_Y(y)$ is the derivative of the CDF with respect to $y$ :

$f_Y(y) = \frac{d}{dy} F_Y(y) = \frac{d}{dy} \sqrt{y}$

$f_Y(y) = \frac{1}{2\sqrt{y}}$

Final Result

The PDF of $Y = X^2$ is:

$f_Y(y) = \begin{cases} \frac{1}{2\sqrt{y}} & \text{if } 0 < y \leq 1 \\ 0 & \text{otherwise} \end{cases}$

Problem Set 3

Mean and variance of transformed variables

Moments (raw and central)

Even/odd moment properties

Moment calculation via PDF

Use of symmetry

Relation between moments and variance

Practice Set 4

Joint distribution functions

Joint PDF and marginal densities

Independence of random variables

Conditional densities

Expectation of functions of two variables

Covariance and correlation

Problem Set 4

Distribution of the sum of random variables
Convolution of PDFs
Sum of independent random variables
Sum of uniform variables
Sum of exponential variables
Mean and variance of sums
Linear combinations of random variables