6. Probability Modeling

Probability Basics — Quiz

Test your understanding of probability basics with 5 practice questions.

Practice Questions

Question 1

A continuous random variable $X$ has a probability density function (PDF) given by $f(x) = cx(2-x)$ for $0 \le x \le 2$ and $f(x) = 0$ otherwise. What is the value of the constant $c$ that makes $f(x)$ a valid PDF?

Question 2

Let $X$ be a discrete random variable with probability mass function $P(X=x) = \frac{k}{x}$ for $x = 1, 2, 3$. What is the value of the constant $k$?

Question 3

If $X$ and $Y$ are independent random variables with $E[X] = 2$, $Var(X) = 3$, $E[Y] = 5$, and $Var(Y) = 4$, what is $Var(3X - 2Y)$?

Question 4

Consider a discrete random variable $X$ with $P(X=1) = 0.2$, $P(X=2) = 0.5$, and $P(X=3) = 0.3$. What is the expected value of $X^2$, i.e., $E[X^2]$?

Question 5

Let $X$ be a continuous random variable with a cumulative distribution function (CDF) given by $F(x) = 1 - e^{-3x}$ for $x \ge 0$ and $F(x) = 0$ for $x < 0$. What is the probability density function (PDF) $f(x)$?