W04 - HW Solutions

Bernoulli Process

04-02 - PMF and CDF: number of heads in five flips

Let count the number of heads resulting from five flips of a coin.

Write complete formulas (using cases) for the PMF and CDF.

Solution

1. & Identify distribution.
   • We have that , since there are trials and .
2. && Use binomial distribution formula to get PMF.
3. & Find CDF.
   • Since , we have
4. & Write out explicit values of CDF.

04-03 - Rolling until a six

A fair die is rolled until a six comes up.

What are the odds that it takes at least 10 rolls? Use a geometric random variable.

Solution

1. & Define random variables.

   • Let .
   • We wish to find .
   • For all , , the first trials result in failure, and the trial is a success.

2. && Compute probability.

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

04-04 - Intersection accidents

Suppose that the odds of an accident occurring on any given day at the intersection of Ivy and Emmett is 0.05.

What are the odds of the first accident occurring after day 4 and by day 10?

(You have calculated the answer before. This time, rework the problem in terms of an appropriate distribution type.)

Solution

1. & Define random variables.
   • Let .
   • We wish to find .
2. & Compute using the formula for a geometric distribution.

04-05 - Components of a car

A very strange car with components will drive if at least half of its components work. Each component will work with the same probability , independently of the others.

For what values of is a car with more likely to drive than a car with ?

(Start by defining a random variable that counts the number of working components.)

Solution

1. & Define random variables.
   • Let represent the car with three components.
   • Let represent the car with five components.
2. && Find probabilities that both cars work.
   • For the three-component car, we want , so
   • For the five-component car, we want , so
3. && Find when .
   • Solve the inequality for .

04-06 - Geometric distribution is memoryless

Suppose that .

Derive this equation: Interpret the equation.

Solution

1. & Set up conditional probability formula.
2. && Find formulas for numerator and denominator.
3. & Plug in values into initial formula

04-07 - Binomial ratios

Suppose .

• Find the value of that maximizes . Do this by studying the successive ratios .
• Use these ratios to compute as a sum of 5 terms without using factorials. Do this by computing directly, and then writing a recursive algorithm that determines in terms of .

Solution

1. & Find formula for ratio .
2. && Interpret ratio.
   • We want , so .
   • Solving for , we get .
   • Since is an integer, the is maximized when .
3. & Compute directly.
   • Based on the figure, and .
4. & Use ratio to solve for successive terms.
5. & Add up probabilities.

04-08 - Prize on the Mall

A booth on the Mall is running a secret prize game, in which the passerby wearing a hat wins 1,000$.

Passersby wear hats independently of each other and with probability 20.

Let be a random variable counting how many passersby pass by before a winner is found.

• (a) What is the name of the distribution of ? What are the parameters?
• (b) What is the probability that the passerby wins the prize?
• (c) What is the probability that at least 7 passersby are needed before a winner is found?

Solution (a)

1. & Identify the distribution.
   • follows a negative binomial distribution with parameters and .

(b)

1. & Compute .

(c)

1. & Compute .
   • We know that the minimum number of passersby before a winner is declared is .
   • Therefore, .

Expectation and variance

04-09 - Students and buses expect different crowding.

Bus One has students, Bus Two has , Bus Three has , and Bus Four has .

• Let measure the number of students on a given random student’s bus.
• Let measure the number of students on a given random driver’s bus.

Compute and . Are they different? Why or why not?

Solution

1. && Compute .
   • There are total students.
   • Let be the probability that Bus is selected, where .
   • Note that .
2. && Compute .
   • Let be the probability that Bus is selected.
   • Since it’s based off the drivers, for all .
3. & Interpret solution.
   • , because the probability that bus was selected in both scenarios varied.

04-10 - Insurance expected payout

A car insurance analytics team estimates that the cost of repairs per accident is uniformly distributed between 100$1500$500$ deductible and covers all costs above the deductible.

How much is the expected payout per accident?

Solution

1. && Find PDF of .
   • If , insurance covers 0$.
   • If , then the insurance covers dollars.
2. & Integrate to find .
   • Since the cost of repairs in uniformly distributed, we have , .

04-11 - Expectation, variance of geometric variable

Derive formulas for and given .

Solution

1. && State the pmf of a geometric random variable.
2. && Use formula for expectation to find .
3. &&& Apply hint.
   • We have that .
   • Differentiating both sides yields .
   • Note that here, , so
4. && Find expression , and note that .
   • Applying the hint, we have .
   • Using the linearity of expectation, we can write this as .
5. &&& Find and
   • First, note that the second derivative of is .
   • Thus, .
6. & Find .