2. Probability Theory

Transformations — Quiz

Test your understanding of transformations with 5 practice questions.

Practice Questions

Question 1

If $X$ is a continuous random variable with probability density function $f_X(x)$, and $Y = g(X)$ is a transformation, what is the formula for the probability density function of $Y$, $f_Y(y)$, using the change-of-variable technique, assuming $g(x)$ is a monotonic function?

Question 2

Consider two independent continuous random variables $X_1$ and $X_2$. If we define two new random variables $Y_1 = g_1(X_1, X_2)$ and $Y_2 = g_2(X_1, X_2)$, what is the general approach to find their joint PDF $f_{Y_1, Y_2}(y_1, y_2)$?

Question 3

If $X$ is a random variable with a PDF $f_X(x) = 2x$ for $0 \le x \le 1$, and $Y = X^2$, what is the cumulative distribution function (CDF) of $Y$, $F_Y(y)$?

Question 4

Given two independent exponential random variables $X_1 \sim Exp(\lambda)$ and $X_2 \sim Exp(\lambda)$, what is the distribution of their sum $Z = X_1 + X_2$?

Question 5

If $X$ is a continuous random variable and $Y = g(X)$ is a transformation, and $g(x)$ is a strictly decreasing function, how does the change-of-variable formula adapt?