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Quiz questions

1.

Valid PDFs Which of the following are valid probability density functions?
(1) $f_{X}(x)=\left\{\begin{array}[]{cl}2&0<x<2\\ 0&\mbox{otherwise}\end{array}\right.$
(2) $f_{X}(x)=\left\{\begin{array}[]{cl}2x/3&-1<x<2\\ 0&\mbox{otherwise}\end{array}\right.$
(3) $f_{X}(x)=\left\{\begin{array}[]{cl}x\exp(-x)&0<x<\infty\\ 0&\mbox{otherwise}\end{array}\right.$

A: All of them B: (2) only C: (2) and (3) D: (3) only E: None of them
2.

Vaild CDFs Which of the following are valid cumulative distribution functions?
(1) $F_{X}(x)=\left\{\begin{array}[]{cl}x&0<x<1\\ 0&\mbox{otherwise}\end{array}\right.$

(2) $F_{X}(x)=\left\{\begin{array}[]{cl}0&x\leq-3\\ 1/2+x^{3}/54&-3<x<3\\ 1&x\geq 3\end{array}\right.$

(3) $F_{X}(x)=\left\{\begin{array}[]{cl}\lambda\exp(-\lambda x)&x>0\\ 0&\mbox{otherwise}\end{array}\right.$

A: All of them B: (2) only C: (2) and (3) D: (3) only E: None of them
3.

Inverse gamma Let $Y\sim\mbox{Gamma}(\alpha,\beta)$ with $\alpha>2$ . What is $E(1/Y)$ ?
A: $\frac{\beta}{\alpha}$ , B: $\frac{\beta}{\alpha-1}$ C: $\frac{\alpha}{\beta}$ D: $\frac{1}{\beta\alpha}$ E: $\frac{1}{\beta(\alpha-1)}$
4.

Sampling Bernoulli Which command will provide you with 30 samples from a Bernoulli distribution with probability 0.8 of observing a 1?

rbern(30,0.8)

pbern(30,0.8)

rbinom(30,1,0.8)

pbinom(30,1,0.8)

runif(30)<0.8
5.

Gamma probability What is the probability that a ${\rm Gamma}(10,5)$ random variable lies between 2 and 4? (A) 0.4125, (B) 0.4529, (C) 0.5013, (D) 0.5612, (E) 0.6210