W06 Homework

Due date: Wednesday 2/18, 11:59pm

Normal distribution

01

06

Gaussian basics

Find the probability that one observation of a Gaussian variable will yield a value within 1.5 standard deviations of the mean.

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Solution

Solutions---5180-06

02

07

Morning commute time

My morning commute time is normally distributed, with a mean of 14 minutes, and a standard deviation of 4 minutes. I leave for work every morning at 8:45am and need to arrive by 9:00am.

(a) On any given day, what is the probability that I am late?

(b) On any given day, what is the probability that I reach before 8:55am?

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Solution

Solutions---5180-07

03

04

Normal distribution - test scores

In a large probability theory exam, the scores are normally distributed with a mean of 75 and a standard deviation of 10.

(a) What is the probability that a student scored between 70 and 80

(b) What is the lowest score a student can achieve to be in the top 5%?

(c) What score corresponds to the 25th percentile?

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Solution

Solutions---5180-04

04

03

Generalized normal

Let $X$ be generalized normal variable with $\mu =3$ and $\sigma =2$. Using a chart of $\mathrm{\Phi }$ values, find:

(a) $P[2<X<6]$

(b) $c$ such that $F_X(c)=0.67$

(c) $E[X^2]$ (Hint: Use $\mu$ and $\sigma$ to avoid integration.)

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Solution

Solutions---5180-03

Review problems

05

02

Insurance expected payout

A car insurance analytics team estimates that the cost of repairs per accident is uniformly distributed between $100 and $1500. The manager wants to offer customers a policy that has a $500 deductible and covers all costs above the deductible.

How much is the expected payout per accident?

(Hint: Graph the PDF for the cost of repairs $X$; write a formula for the payout in terms of $X$ using cases; then integrate.)

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Solution

Solutions---5120-02

06

05

CDF of derived variable

Suppose $X$ is a continuous $\mathrm{Unif}[\mathrm{1,4}]$ random variable. Let $Y=|X-2|$. Find the CDF of $Y$.

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Solution

Solutions---5150-05

07

06

Octane revenue

The owner of a small gas station has his 1,500 gallon tank of 93-octane gas filled up once at the beginning of each week. The random variable $X$ is the amount of 93-octane the station sells in one week (in thousands of gallons). The PDF of X is shown below.

$$f_X(x)=\{\begin{array}{cc}x& 0\le x\le 1\\ 1& 1<x\le 1.5\end{array}$$

Assuming the station consistently charges $3.00 per gallon for 93-octane and pays $2,000 for the weekly fill-up, find the CDF of $W=3X-2$, the profit the station makes from the 93-octane in a week.

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Solution

Solutions---5150-06