Problems

01

Constants in PDF from expectation

Suppose $X$ has PDF given by:

$$f_X(x)=\{\begin{array}{cc}a+b{x}^2& 0\le x\le 1\\ 0& \text{otherwise}\end{array}$$

Suppose $E[X]=\frac{7}{10}$. Find the only possible values for $a$ and $b$. Then find $\mathrm{Var}[X]$.

02

Variance: Direct integral formula

Suppose $X$ has PDF given by: $f_X(x)=\{\begin{array}{cc}3e^{-3x}& x\ge 0\\ 0& \text{otherwise}\end{array}$

Find $\mathrm{Var}[X]$ directly using the integral formula.

03

PDF of derived variable for $E[X]$ and $\mathrm{Var}[X]$

Suppose the PDF of an RV is given by:

$$f_X(x)=\{\begin{array}{cc}\frac{3}{4}x(2-x)& 0\le x\le 2\\ 0& \text{otherwise}\end{array}$$

(a) Find $E[X]$ using the integral formula.

(b) Find $f_{X^2}(x)$, the PDF of $X^2$ (by calculating the CDF first).

(c) Find $E[X^2]$ using $f_{X^2}(x)$.

(d) Find $\mathrm{Var}[X]$ using results of (a) and (c).

04

Random point in ${[\mathrm{0,1}]}^2$, PDF of $X$ and $X^2$

A random point is chosen in the unit square $[\mathrm{0,1}]\times [\mathrm{0,1}]$.

Outcomes are points $(x,y)$ in this square. Events are regions in the square. The probability of a region $A$ is the area of $A$.

Define the random variable $X$ by $X(x,y)=x$. This is just the $x$-coordinate of the random point. Then the random variable $X^2$ is given by $X^2(x,y)=x^2$. This is just the squared $x$-coordinate of the random point.

(a) Describe the PDF of $X$.

(b) Describe the PDF of $X^2$.

05

CDF of derived variable

Suppose $X$ is a continuous $\mathrm{Unif}[\mathrm{1,4}]$ random variable. Let $Y=|X-2|$. Find the CDF of $Y$.

06

Octane revenue

The owner of a small gas station has his 1,500 gallon tank of 93-octane gas filled up once at the beginning of each week. The random variable $X$ is the amount of 93-octane the station sells in one week (in thousands of gallons). The PDF of X is shown below.

$$f_X(x)=\{\begin{array}{cc}x& 0\le x\le 1\\ 1& 1<x\le 1.5\end{array}$$

Assuming the station consistently charges $3.00 per gallon for 93-octane and pays $2,000 for the weekly fill-up, find the CDF of $W=3X-2$, the profit the station makes from the 93-octane in a week.

07

Square root

Suppose that $X\sim \mathrm{Exp}(1)$.

Let $Y=\sqrt{X}$. Find the PDF of $Y$.

08

Derived random variable

Let $U=\mathrm{Unif}[\mathrm{0,1}]$ and suppose that $X=-\ln (1-U)$.

(a) Compute $F_X(x)$.

(b) Compute $f_X(x)$.

(c) Compute $E[X]$.