Probability density and cumulative distribution functions Quiz

1. $\mathrm{f}(x)=kx$ for $0\leq x\leq2$ and $0$ otherwise. Find $k$.

• A. $\frac12$
• B. $2$
• C. $\frac14$
• D. $1$

2. $\mathrm{F}(x)=\frac{x^2}{16}$ for $0\leq x\leq4$. Find $\mathrm{P}(X\leq3)$.

• A. $\frac{3}{16}$
• B. $\frac{9}{16}$
• C. $\frac34$
• D. $\frac14$

3. $\mathrm{F}(x)=\frac{x^3}{8}$ for $0\leq x\leq2$. Find $\mathrm{f}(x)$ in this range.

• A. $\frac{x^4}{32}$
• B. $\frac{x^2}{8}$
• C. $\frac{3x^2}{8}$
• D. $3x^2$

4. Which expression equals $\mathrm{P}(a<X\leq b)$?

• A. $\mathrm{f}(b)-\mathrm{f}(a)$
• B. $\mathrm{F}(a)-\mathrm{F}(b)$
• C. $\int_a^b\mathrm{F}(x)\,\mathrm{d}x$
• D. $\mathrm{F}(b)-\mathrm{F}(a)$

5. $\mathrm{f}(x)=2x$ for $0\leq x\leq1$. Find $\mathrm{P}(X>0.5)$.

• A. $0.25$
• B. $0.75$
• C. $0.5$
• D. $1$