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Probability Review Questions

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Question 1 The following five (5) situations require the application of one of the common discrete or common continuous distributions we learned about. All you have to do for each of these problems is identify the correct probability mass function or probability density function name. DO NOT SOLVE THE PROBLEM. There is only one correct answer for each problem and each answer may (or may not) be used more than once. Your choices of possible answers include the following:

Bernoulli pmf Binomial pmf Poisson pmf Exponential pdf Uniform pdf

1a. Ten independent rocket missions to the Mars are planned by NASA. The probability that a rocket makes it to Mars successfully is .95. What is the probability that at least 8 of these rockets make it to Mars? What pmf or pdf would you use to solve this problem ?

1b. A vehicle is randomly located on a 400-mile stretch of I-10 between Phoenix and Los Angeles. The location of this vehicle, X, has CDF F(x) = 1/400 for 0 < x < 400. What pmf or pdf does X have?

1c. Pedestrians cross at an intersection. T, their inter-arrival times (times between consecutive arrivals) has this CDF: F(t) = 1 - e-t for t > 0. What pdf or pmf does T have?

1d. A certain genetic test can determine if a person is a carrier of breast cancer gene or not. X = 1 with probability p if a person is a carrier and X = 0 with probability 1-p if a person is not a carrier. What pmf or pdf does X have?

1e. An intersection has 30 cars/min pass through it. The probability that x = 25 pass through this intersection in one minute is

What is the name of the pdf or pmf of X?

Question 2. The weight of small motor is modeled as a uniformly distributed random variable between 10 and 11 lbs. For ease, call this weight W.

a. Graph the pdf of W. Label your axes and show any important numerical values on them.

b. Graph the CDF of W. Label your axes and show any important numerical values on them.

c. Twenty (20) of these motors will be placed into a shipping box. If the weight of a box exceeds 212 lbs, there will be an extra shipping charge of $20.00 applied. What is the probability that the shipping charge is applied?

Bonus : What is the probability that the weight of a randomly selected box is 11 lbs?

Question 3 Bob is responsible for inspecting Product A and Product B. Both products arrive at his inspection station independently of one another. Product A arrives to his station at a rate of 3 every 30 minutes according to a Poisson pmf. Product B arrives to his station at a rate of 5 every 30 minutes also according to a Poisson pmf.

a. What is the mean number of Product A that Bob inspects in one hour?

b. What is the probability that Bob inspects more than 8 of Product B in the next hour? You can leave your answer in term (summation) form. No need to crunch the number.

Question 4 At a restaurant, daily demand (D) for a specific dish (e.g., Peking Duck) has the following associated probabilities. (There has never been more than three requests for this dish):

P(D = 0) = .10 P(D = 1) = .40 P(D = 2) = .30 P(D = 3) = .20

a. What is the probability that the demand is 4?

b. What is the expected demand?

c. What is the variance of the demand?

Question 5 Random variable X has p.d.f. for 2 < x < m

a. What is the value of m?

b. What is P(X =2) + P(X = 5)?

c. What is the probability that X is larger than 5?

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The expert examines the probability for real numbers.

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Please see the attached solutions. If anything is unclear, please let me know and I'll be happy to explain.

Regarding 2c)
The only other way I can think of doing it is using the Central Limit Theorem as you mentioned. However, Central ...

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