Probability and Statistics Problems: A Comprehensive Set
1. Four-Digit Lock
(a) How many combinations are made with the first digit 1 or 2 and digits repeat?
(b) If the first digit is 1 or 2, what is the probability of all digits being unique?
2. Probability Mass Function (PMF) Table
(a) Present the cumulative distribution function (CDF) of X.
| 0 | 1 | 2 | 3 | 4 | 5 | |
|---|---|---|---|---|---|---|
| PMF | 0.1 | 0.1 | 0.25 | 0.4 | 0.1 | 0.05 |
(b) Compute E(X) and STD(X).
(c) If a day pass is $5, how much is spent on the trolley?
3. Flight and Luggage Problem
Leah is flying from Boston to Denver with a connection in Chicago. The probability her first flight leaves on time is 0.15. If the flight is on time, the probability that her luggage will make the connecting flight in Chicago is 0.95, but if the first flight is delayed, the probability that the luggage will make it is only 0.65. Let A be the event that the first flight leaves on time, and let B be the event that the luggage will make the connecting flight.
(a) Are events A & B independent? Explain.
(b) What is the probability that her luggage arrives in Denver with her?
(c) If the luggage does not arrive in Denver with her, what is the probability the first flight did not leave on time?
4. Probability Distribution Identification
State the probability distribution with parameter(s) best suited for each situation. Present the mean and variance.
(a) If a single bit (0 or 1) is transmitted over a noisy communications channel, it has a probability of 0.1 of being incorrectly transmitted. To overcome this problem, the bit is transmitted 5 times. A decoder at the receiving end, called a majority decoder, decides that the correct message is that carried by a majority of the received bits. Under a simple noise model, each bit is independently subject to being corrupted with the same probability 0.1. Consider the distribution of the number of bits in error, X.
(b) In pick-up basketball games, the team to start with the ball is often decided by shooting free throws. The first team to make a free throw gets the ball. If the shooter for a team hits free throws on 80% of shots, consider the distribution of the number of free throws needed until one is made.
(c) The number of calls arriving at an office between 9 and 10 am averages about 30. Consider the distribution of the number of calls on a typical day between 9:45 and 10 am.
(d) A fair coin is tossed until heads appears 4 times. Consider the distribution of the number of tosses.
5. Blood Type Distribution
A study of the Chinese population of Hong Kong in 1937 determined blood types were of 3 types: M, N, and MN, where M and N are erythrocyte antigens. The probabilities in which these three occur in the population are 0.33, 0.18, and 0.49.
(a) Which probability distribution best models the distribution of blood types?
(b) In a sample of 1029 people, how many on average will have each of the three blood types?
(c) In a sample of 1029 people, what is the variance of the number of people with each of the three blood types?
(d) What is the probability that in a group of 12 independent people, the blood types are distributed evenly (4 people have each of the 3 blood types)?
6. Capture/Recapture Method
The so-called capture/recapture method is sometimes used to estimate the size of a wildlife population. Suppose that 10 animals are captured, tagged, and released. On a later occasion, 20 animals are captured, and it is found that 4 of them are tagged.
(a) If the population size is estimated at 50, what is the probability of capturing 4 tagged animals in this sample of 20?
(b) If the population size is 50, what is the expected number of tagged animals captured and the standard deviation of this number, in this sample of 20?
(c) Based on your answers in (a) and (b), is the estimated population size of 50 plausible?
(d) How might you use the recapture probability distribution from (a) to guess the population size?
7. Continuous Distribution Problems
Solve the following continuous distribution problems.
(a) An athlete completed an 800m race in 150 seconds. The distribution of 800m race times followed a bell curve, with a mean of 165 seconds and a standard deviation of 7. The same athlete also completed a swim, finishing in 12.25 minutes. The distribution of swim times also followed a bell curve, with a mean of 15 minutes and a standard deviation of 1.5 minutes. In which event does the athlete have a better standing relative to the other competitors in the event?
(b) It is 9:00 PM. The time until Joe receives his next text message has an exponential distribution with a mean of 5 minutes. Find the probability that the next text arrives between 9:07 and 9:10 PM.
(c) The minimum time for firefighters at a certain station to reach the scene of a fire is 3 minutes. The actual time is 3 + X, where the random variable X has a gamma distribution with U = 4 minutes and O = 2 minutes. Find a and b.
(d) A density function is proportional to f(x) = x^2(1-x)^3 for 0 < x < 1. Find the mean and variance of the distribution.
(e) A PE instructor studied the vertical distances that students in a 9th-grade class could jump. The instructor found that these distances follow a Weibull distribution with a mean of 10 inches and a standard deviation of 3 inches. What range of distances constitutes the middle 50% of the probability density?
(f) In a population, suppose personal income above $15,000 (units are $10K) has a Pareto distribution. The probability that a randomly chosen individual has income greater than $60,000 is 0.0825. Find a.
8. Linear Hazard Rate Function
Assume a random variable has a linear hazard rate function; that is, h(t) = a + bt for constants A and B.
(a) Find the distribution function for this random variable.
(b) Find the density function for this random variable.
(c) When A = 0 in the linear hazard rate function, the random variable is said to have a Rayleigh distribution. The Rayleigh distribution is a failure time distribution commonly used in actuarial science. Assume X is the failure time, in decades, of a timing belt on a car, following a Rayleigh distribution with b=5. What is the probability the device fails in less than a decade?
9. Pareto Distribution for City Populations
In Newman (2005), the population of U.S. cities is modeled by a Pareto distribution with a mean of 7 (units are 10K people) and a standard deviation of 8.5. Use the method of moments to find the two parameters m and a.
10. Restaurant Waiting Times
Customers at a popular restaurant are waiting to be served. Waiting times are independent and exponentially distributed with a mean 1/λ = 30 minutes. If 30 customers are waiting, what is the probability that their average wait time is less than 25 minutes?
11. Credit Card Bill Payments
A survey of 1500 credit card customers showed that 510 pay their bills in full each month. Form an approximate 95% confidence interval for the fraction of customers in the population who pay their bills in full each month.