Homework 5: Practice Exam
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 | 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 |