Homework 5: Practice Exam


March 11, 2023

Problem 1

Let X and Y be two random variables that have the following joint distribution function:

f(x,y)=\left\{\begin{array}{cc} e^{-x/2}ye^{-y²} & x>0;y>0\\ 0 & \mathrm{otherwise} \end{array} \right.

What is the marginal distribution of Y?

Problem 2

Show that the moment generating function for a random variable X\sim Gamma(\alpha,\beta) is \left(1-\beta t\right)^{-\alpha}.

Problem 3

Let Y have the following density function:

f(y) = \left\{\begin{array}{cc} k[1-(x-3)^2] & 2\le x \le 4 \\ 0 & \mathrm{otherwise} \end{array}\right.

Find the value of k that makes this a valid distribution.

Problem 4

Let X be a random variable following a logistic distribution with parameters \mu and s. What is E(X^2)? Note, you do not need to derive it.

Problem 5

Let X have the following density function:

f(x) = \left\{\begin{array}{cc} x/25 & 0\le x \le 5 \\ 2/5 - x/25 & 5 < x \le 10 \\ 0 & \mathrm{otherwise} \end{array}\right.

Find the P(4\le X\le 7).

Problem 6

Show that the moment generating function for a random variable X\sim N(\mu,\sigma^2) is e^{\mu t+ \frac{t^2}{2}\sigma^2}.

Discrete Distributions

Distribution \theta PMF E(X) Var(X) MGF Support
Bernoulli p p p p(1-p) 1-p+pe^t X = 0,1
Binomial n,p (^n_x)p^x(1-p)^{n-x} np np(1-p) (1-p+pe^t)^n X = 0,1,\ldots,n
Poisson \lambda \frac{e^{-\lambda}\lambda^x}{x!} \lambda \lambda e^{\lambda(e^t-1)} X = 0,1,\ldots,\infty
Geometric p p(1-p)^{x-1} \frac{1}{p} \frac{1-p}{p²} \frac{pe^t}{1-(1-p)e^t} X=1,2,\ldots,\infty
Negative Binomial r,p (^{x-1}_{r-1})p^{r-1}(1-p)^{x-r} \frac{pr}{1-p} \frac{(1-p)r}{p^2} \left(\frac{1-p}{1-pe^t}\right)^n X=0,1,\ldots,\infty

Continuous Distributions

Distribution \theta PDF E(X) Var(X) MGF Support
Uniform a,b \frac{1}{b-a} \frac{a+b}{2} \frac{(b-a)^2}{12} \frac{e^{tb}-e^{ta}}{t(b-a)} a \le X \le b
Normal \mu,\sigma^2 \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \mu \sigma^2 e^{\mu t+ \frac{t^2}{2}\sigma^2} -\infty \le X \le \infty
Exponential \lambda \lambda e^{-x\lambda} 1/\lambda 1/\lambda^2 \frac{\lambda}{\lambda-t} 0 \le X
\chi^2 k \frac{1}{2^{k/2}\Gamma(k/2)}x^{k/2-1}e^{-x/2} k 2k (1-2t)^{-k/2} 0 \le X
Gamma \alpha,\beta \frac{1}{\beta^{\alpha}\Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta} \alpha\beta \alpha\beta^2 \left(1-\beta t\right)^{-\alpha} 0 \le X
Beta \alpha,\beta \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}x^{\alpha-1}(1-x)^{\beta} \frac{\alpha}{\alpha+\beta} \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} 0\le X \le 1
Laplace \mu,b \frac{1}{2b}e^{-\frac{|x-\mu|}{b}} \mu 2b^2 \frac{e^{\mu t}}{1-b^2t^2} -\infty\le X\le \infty
Logistic \mu,s \frac{e^{\frac{-(x-\mu)}{s}}}{s\left(1+e^{\frac{-(x-\mu)}{s}}\right)^2} \mu \frac{s^2\pi^2}{3} e^{\mu t}\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} -\infty\le X\le \infty