Probability density and cumulative distribution functions Quiz

1. $f(x)=kx$ for $0\le x\le4$, and 0 otherwise. Find $k$.

• A. $\frac14$
• B. $\frac18$
• C. $\frac1{16}$
• D. $\frac12$

2. $F(x)=\frac{x^2}{9}$ for $0\le x\le3$. Find $f(x)$ on this interval.

• A. $\frac{x}{9}$
• B. $\frac{x^3}{27}$
• C. $2x$
• D. $\frac{2x}{9}$

3. $f(x)=\frac x2$ for $0\le x\le2$. Find $P(X<1)$.

• A. $\frac14$
• B. $\frac12$
• C. $\frac18$
• D. $\frac34$

4. Which statement must be true for a probability density function?

• A. $f(x)\le1$ for all $x$.
• B. $f(x)$ is increasing.
• C. $f(x)\ge0$ for all $x$ and the total area under the curve is 1.
• D. The total area under the curve is 0.

5. For a continuous random variable $X$, what is $P(X=2)$?

• A. $f(2)$
• B. $0$
• C. $F(2)$
• D. $\frac12$