W09 Homework A

Due date: Thursday 3/12, 11:59pm

Functions on two random variables

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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 $W$.

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Solution

Solutions---5210-03

02

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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 $W=\mathrm{Min}(X,Y)$

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Solution

Solutions---5210-02

03

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PDF of sum of uniforms

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

Find the PDF of $W$.

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Solution

Solutions---5220-05

04

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PDF of sum from joint PDF

Suppose the joint PDF of $X$ and $Y$ is given by:

$$f_{X,Y}=\{\begin{array}{cc}{\displaystyle \frac{8}{81}}xy& 0\le y\le x\le 3\\ 0& \text{otherwise}\end{array}$$

Find the PDF of $X+Y$.

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Solution

Solutions---5220-01

Review

Derived RV

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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 $W=\ln X$.

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Solution

Solutions---5210-01

Joint distributions

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Finish a PMF table - Strange families

Suppose that 15 percent of the families in a strange community have no children, 20 percent have 1 child, 35 percent have 2 children, and 30 percent have 3 children. Assume the odds of a child being a boy or a girl are equal.

If a family is chosen at random from this community, then $B$, the number of boys, and $G$, the number of girls, in this family will have the joint PMF partially shown in this table:

$B\downarrow G\to$	0	1	2	3	$P_B(i):$
0	0.15	0.10	?	0.0375	0.3750
1	0.10	0.175	?	0.00	0.3875
2	?	?	0.00	0.00	0.2000
3	0.0375	0.00	0.00	0.00	0.0375
$P_G(j):$	0.3750	0.3875	0.2000	0.0375	[not used]

(a) Complete the table by finding the missing entries.

(b) What is the probability that “$B$ or $G$ is 1”?

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Solution

Solutions---5190-01

07

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Soft-drink machine

A soft-drink machine has a random amount $Y$ in supply at the beginning of a given day and dispenses a random amount $X$ during the day (with measurements in gallons). It is not resupplied during the day, and therefore $X\le Y$. It has been observed that $X$ and $Y$ have a joint density given by:

$$f_{X,Y}(x,y)=\frac{1}{2},0\le x\le y\le 2$$

(a) Find the probability that the amount of soft-drink dispensed on a given day is greater than 1.5 gallons. (First compute the marginal PDF of $X$.)

(b) Find the probability that the amount of soda remaining in the machine at the end of the day is greater than $1/2$ gallon. That is, find $P[Y-X>1/2]$.

(c) Find the CDF of $W=Y-X$, the amount of soda remaining in the machine at the end of the day.

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Solution

Solutions---5190-10

Independent random variables

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Random point in a triangle

Consider a joint distribution that is uniform over the triangle with vertices $(\mathrm{0,0}),(\mathrm{0,1})$, and $(\mathrm{1,0})$. Suppose a point $(X,Y)$ is chosen at random according to this distribution.

(a) Find the joint PDF $f_{X,Y}$.

(b) Find the marginal PDFs for $X$ and $Y$.

(c) Are $X$ and $Y$ independent?

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Solution

Solutions---5200-01