W11 Homework A

Due date: Friday 3/27, 11:59pm

Conditional distribution

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Conditional density from joint density

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

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

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

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

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Solution

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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 $X$.

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

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.

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Conditional expectation

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Conditional distribution and expectation from joint PMF

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

$$P_{X,Y}(k,\ell )=\{\begin{array}{ll}c& k=1,2,3,4;\ell =1,\dots ,k\\ 0& \text{ otherwise }\end{array}$$

Notice that the possibilities for $\ell$ depend on the choice of $k$.

First, show that $c=1/10$. Then compute:

(a) $P_X$ (b) $P_{Y|X}$ (c) $E[Y\mid X=4]$ (d) $E[Y\mid X]$

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Solution

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