W05 - HW Solutions

01

(1) State CDF of a Poisson distribution.

We know that

We know that .

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(2) Compute limit as .

Note that

02

(1) State PMF of a Poisson distribution.

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(2) Find expectation.

We know that .

03

(a)

Write explanation.

• You use Poisson distribution if events occur randomly and you know the mean number of events that occur within a given interval of time.
• In addition, Poisson distributions are advantageous when describing rare events. Since meteors are a rare occurrence, it makes sense to use a Poisson distribution.

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(b)

Compute probability.

Since , it’s easier to compute the latter.

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(c)

Compute probability.

We know that there are 13 meteors in 4 hours, so we see an average of meteors per hour. Let

We wish to find the probability .

04

(a)

(1) Identify distribution.

Clearly, this scenario follows a binary distribution.

We have a chance that the dime is made of silver.

Since we have 40$ worth of dimes, there are 400 dimes.

Thus, .

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(2) Find formula for probability.

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(b)

(1) Find corresponding Poisson distribution.

Let .

.

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(2) Compute probability.

We have that .

05

(a)

(1) Define random variable that is the Poisson approximation to .

.

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(2) Estimate error.

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(b)

(1) Compute using the binomial distribution.

We have that .

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(2) Compute using the Poisson distribution.

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(3) Compute error.

06

(1) Recall formula for expectation of a continuous random variable.

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(2) Use formula to find an equation relating and .

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(3) Recall that integrating a PDF should yield . Integrate the PDF and find a second equation relating and .

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(4) Solve system of equations for and .

Isolating in the second equation yields .

Plugging this expression into the first equation yields .

Solving for yields 2.4, and thus .

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(5) Compute variance using the formula

Compute .

07

(1) Recall the integral formula for variance.

Use the fact that

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(2) Compute .

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(3) Compute

08

(a)

Compute .

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(b)

(1) Find the CDF of .

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(2) Find the CDF of .

Since is monotone increasing, .

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(3) Find the PDF of by differentiating.

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(c)

Find .

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(d)

Compute .

09

(1) State the PDF of an exponential distribution.

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(2) Compute using the integral formula.

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(3) Compute using the integral formula.

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(4) Compute

10

Recall the memoryless property of exponential distributions.

• Elapsed time has no effect on future events.
• Therefore, the fact that one car is older than the other has no effect on the remaining lifetimes.

Derive conclusions.

• Since both cars have the same remaining lifetime distribution, the probability that either car outlives the other is 0.5.

11

(a)

Compute probability the wait time for a Bundle is at most 1 hr.

Our bounds will be from 0 to 1 since we are only concerned about 1 hour.

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(b)

State the Erlang distribution.

Compute desired probability.

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(c)

State conclusions.

• Clearly, the results disagree. This is because method 1 considers calls coming in at bundles at a time instead of considering 5 discrete calls. Method 2 is more accurate since it considers the rates of individual calls.

12

(a)

Out of trials, we choose of them to be heads. Thus,

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(b)

Out of 7 trials, we choose at least 3 of them to be heads.

Using summation notation, we get

13

(1) Define random variables.

Let .

We wish to find .

For all , , the first trials result in failure, and the trial is a success.

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(2) Compute probability.

Note that the summation is simplified using the formula for a geometric series.

14

(a)

(b)

15

(a)

(b)

16

(a)

(b)