4. Statistics

Continuous Distributions — Quiz

Test your understanding of continuous distributions with 5 practice questions.

Practice Questions

Question 1

A continuous random variable $X$ has a probability density function (PDF) given by $f(x) = \begin{cases} k(x^2+1) & 0 \le x \le 3 \\ 0 & \text{otherwise} \end{cases}$. What is the value of the constant $k$?

Question 2

If $X \sim Exp(\lambda)$, and the median of $X$ is $2 \ln(2)$, what is the rate parameter $\lambda$?

Question 3

A random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$. If $Y = \frac{X - \mu}{\sigma}$, what is the mean and variance of $Y$?

Question 4

According to the Central Limit Theorem, if samples of size $n$ are drawn from a population with mean $\mu$ and standard deviation $\sigma$, what is the standard deviation of the distribution of sample means, $\sigma_{\bar{X}}$?

Question 5

A continuous random variable $X$ has a cumulative distribution function (CDF) given by $F(x) = \begin{cases} 0 & x < 0 \\ 1 - e^{-0.2x} & x \ge 0 \end{cases}$. What is the probability density function (PDF) $f(x)$?