(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Jake leaves home at a random time between 7:30 and 7:55 a.m.

(assume the uniform distribution) and walks to his office. The walk takes 10

minutes. Let T be the amount of time spends in his office between 7:40 and

8:00 a.m.. Find the distribution function F_T of T and draw its graph. Does F_T

have a density?

2. Relevant equations

3. The attempt at a solution

This is what I've come up with so far:

Let X be the number of minutes past 7:30 that he leaves his house.

[tex]T = \begin{cases} 30 - (x + 10) &\text{if } 0\leq x \leq 20 \\ 0 &\text{if } 20<x\leq 25 \end{cases}[/tex]

[tex]F_T(t) = \mathbb{P}[T\leq t] = \begin{cases} \mathbb{P}[20-x\leq t] &\text{if } 0\leq t \leq 20 \\ 0 &\text{if } t<0 \\ 1 &\text{if } t>20\end{cases} [/tex]

[tex] \mathbb{P}[20-x \leq t] = 1 - \mathbb{P}[x<20 -t][/tex]

[tex]= 1 - \int_{0}^{20-t} \frac{1}{25} dx[/tex]

[tex] = \frac{5+t}{25}[/tex]

[tex]\therefore F_T(t) = \mathbb{P}[T\leq t] = \begin{cases} \frac{5+t}{25} &\text{if } 0\leq t \leq 20 \\ 0 &\text{if } t<0 \\ 1 &\text{if } t>20\end{cases}[/tex]

This would imply no density, but that seems plausible given that for [tex]T=0[/tex], [tex] 20<X\leq 25[/tex].

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# Homework Help: Probability - Uniform distribution word problem.

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