W11 Homework B

Due date: Tuesday 3/31, 11:59pm

Conditional distribution

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

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Solution

Solutions---5250-02

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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 $0.3\mathrm{lbs}$.

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Solution

Solutions---5250-04

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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 $Y$.

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Solution

Solutions---5250-05

Conditional expectation

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Conditional distribution and expectation from joint PDF

Suppose that $X$ and $Y$ have the following joint PDF:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}cxy& 0<y<1,0<x<y\\ 0& \text{ otherwise }\end{array}$$

Notice that the range of possibilities for $x$ depends on the value of $y$.

First, show that $c=8$. Then compute:

(a) $f_X$ $$ (b) $f_{Y|X}$ $$ (c) $E[Y|X=0.5]$ $$ (d) $E[Y|X]$

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Solution

Solutions---5260-02