Problems

01

Function on one variable

Suppose the PDF of $X$ is given by:

$$f_X(x)=\{\begin{array}{cc}\frac{2}{3}x& 1\le x\le 2\\ 0& \text{otherwise}\end{array}$$

Find the CDF and PDF of $W=\ln X$.

02

PDF of Min and Max

Suppose $X\sim \mathrm{Exp}(2)$ and $Y\sim \mathrm{Exp}(3)$ and these variables are independent. Find:

(a) The PDF of $W=\mathrm{Max}(X,Y)$

(b) The PDF of $W=\mathrm{Min}(X,Y)$

03

PDF of Min

Let $X$ and $Y$ be independent copies of a $\mathrm{Unif}[\mathrm{0,1}]$ random variable.

Let $W=\mathrm{Min}(X,Y)$. Find the PDF of $W$.

04

Coffee and danish

Joe visits a coffee shop every morning and orders a triple espresso and a cherry danish. Let $X$ represent the wait time for the espresso and $Y$ the wait time for the danish (both in minutes). The joint PDF of $X$ and $Y$ is:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}\frac{1}{50}{e}^{-\frac{1}{5}x-\frac{1}{10}y}& x,y>0\\ 0& \text{ otherwise }\end{array}$$

Suppose that Joe will not sit down until he has both his espresso and his danish. Compute the probability that he will have to wait at least 5 minutes before sitting down, that is, find $P[\mathrm{Max}(X,Y)>5]$.