W05 Homework A

Due date: Thursday 2/12, 11:59pm

Poisson process

01

05

Potholes on the highway

On a certain terrible stretch of highway, the appearance of potholes can be modeled by a Poisson process. Let the RV $X$ denote the distance between successive potholes (measured in miles). The CDF of X is:

$$F_X(x)=\{\begin{array}{cc}0& x\le 0\\ 1-e^{-0.5x}& x>0\end{array}$$

(a) What is the mean number of potholes in a 2-mile stretch of the highway?

(b) What is the probability that there will be at least 2 potholes in a 2-mile stretch of the highway?

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Solution

Solutions---5160-05

02

04

Selling Christmas trees

An online company sells artificial Christmas trees. During the holiday season, the amount of time between sales, $T$, is an exponential random variable with an expected value of 2.5 hours.

(a) Find the probability that the store will sell more than 2 trees in a 1-hour period of time.

(b) Find the probability the time between the sales of two trees will be between 4-5 hours.

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Solution

Solutions---5160-04

Derived random variable

03

01

Constants in PDF from expectation

Suppose $X$ has PDF given by:

$$f_X(x)=\{\begin{array}{cc}a+b{x}^2& 0\le x\le 1\\ 0& \text{otherwise}\end{array}$$

Suppose $E[X]=\frac{7}{10}$. Find the only possible values for $a$ and $b$. Then find $\mathrm{Var}[X]$.

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Solution

Solutions---5150-01

04

02

Variance: Direct integral formula

Suppose $X$ has PDF given by: $f_X(x)=\{\begin{array}{cc}3e^{-3x}& x\ge 0\\ 0& \text{otherwise}\end{array}$

Find $\mathrm{Var}[X]$ directly using the integral formula.

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Solution

Solutions---5150-02

Review problems

05

05

Binomial - Repeated coin flips

A coin is flipped 7 times and the sequence of results recorded as an outcome.

(a) How many possible outcomes have exactly 3 heads?

(b) How many possible outcomes have at least 3 heads?

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Solution

Solutions---5070-05