Problems

01

Conditional density from joint density

Suppose that $X$ and $Y$ have joint probability density given by:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}{\displaystyle \frac{12}{5}x(2-x-y)}& x,y\in [\mathrm{0,1}]\\ & \\ 0& \text{ otherwise }\end{array}$$

(a) Compute $f_{X|Y}(x|y)$, for $y\in [\mathrm{0,1}]$.

(b) Compute $P[X>1/2|Y=y]$.

02

From conditional to joint, and back again

Suppose we have the following data about random variables $X$ and $Y$:

$$\begin{array}{c}{f}_X(x)=\{\begin{array}{cc}3x^2& 0\le x\le 1\\ 0& \text{ otherwise }\end{array}\\ \\ f_{Y|X}(y|x)=\{\begin{array}{cc}2y/x^2& 0\le y\le x\\ 0& \text{ otherwise }\end{array}\end{array}$$

(a) Find the joint distribution $f_{X,Y}(x,y)$.

(b) Find $f_{X|Y}(x|y)$.

03

Time till recharge

Let $X$ denote the amount of time (in hours) that a battery on a solar calculator will operate properly before needing to be recharged by exposure to light. The function below is the PDF of $X$.

$$f(x)=\{\begin{array}{cc}{\displaystyle \frac{50}{6}}{x}^{-3}& 2<x<10\\ & \\ 0& \text{ otherwise }\end{array}$$

Suppose that a calculator has already been in use for 5 hours. Find the probability it will operate properly for at least another 2 hours.

04

Cashews in a can

A nut company markets cans of mixed nuts containing almonds, cashews, and peanuts. Suppose the net weight of each can is exactly $1\mathrm{lb}$, but the weight contribution of each type of nut is random. Let $X$ be the weight of almonds in a selected can and $Y$ the weight of cashews. The joint PDF of $X$ and $Y$ is given below:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}24xy& x,y\in [\mathrm{0,1}],x+y\le 1\\ 0& \text{otherwise}\end{array}$$

Suppose that the weight of cashews in a particular can is $0.5\mathrm{lbs}$. Calculate the probability that the weight of almonds in this can is more than $0.3\mathrm{lbs}$.

05

New and returning customers

A sales representative will randomly select and call 2 customers. The representative’s goal is to get each customer to complete a satisfaction survey. Each of these customers is categorized as “new” or “returning.” 70% of customers are new and 30% are returning. Let $X$ be the number of new customers that are called.

(a) Construct the PMF of $X$, $P_X(x)$.

The probability of any new customer completing the survey is 0.15, and the probability of any returning customer completing the survey is 0.20. (Customers operate independently.) Let $Y$ be the number of new customers that complete the survey.

(b) Construct $P_{Y|X}(y|0)$, $P_{Y|X}(y|1)$, and $P_{Y|X}(y|2)$. (You should construct 3 separate PMFs.)

(c) Construct the joint PMF of $X$ and $Y$.