Question 1

The conditions for both the hypergeometric and the binomial random variables require that each trial results in one of two outcomes.

Accepted Answer

Both the hypergeometric and binomial distributions model situations where each trial has only two possible outcomes, often referred to as "success" and "failure." The key difference lies in the dependency of trials in hypergeometric distributions (without replacement, affecting probabilities of subsequent trials) versus the independence of trials in binomial distributions (with replacement, keeping probabilities constant).

Question 2

The hypergeometric random variable x counts the number of successes in the draw of 3 elements from a set of 8 elements containing 4 successes. List the possible values of x.

Accepted Answer

The answer of The hypergeometric random variable x counts the...

Question 3

Given that x is a hypergeometric random variable with N = 15, n = 6, and r = 10, compute P(x = 0).

Accepted Answer

P(x = 0) = (10 choose 0) * (5 choose 6) / (15 choose 6) = 0.001

Question 4

Suppose that 4 out of 12 liver transplants done at a hospital will fail within a year. Consider a random sample of 3 of these 12 patients. What is the probability that all 3 patients will result in failed transplants?

Accepted Answer

The probability of a failed transplant is 4/12 = 1/3. The probability of all 3 patients failing is (1/3)^3 = 1/27 = 0.037.

Question 5

Given that x is a hypergeometric random variable, compute p(x) for N = 6, n = 3, r = 3, and x = 1.

Accepted Answer

The answer of Given that x is a hypergeometric random...

Question 6

An alarm company reports that the number of alarms sent to their monitoring center from customers owning their system follow a Poisson distribution with λ = 4.7 alarms per year. Identify the mean and standard deviation for this distribution.

Accepted Answer

The mean of a Poisson distribution is equal to the lambda parameter, which is 4.7 in this case. The standard deviation of a Poisson distribution is equal to the square root of the lambda parameter, which is 2.17 in this case.

Question 7

Given that x is a hypergeometric random variable with N = 10, n = 3, and r = 6, compute P(x = 0).

Accepted Answer

The probability P(x = 0) for a hypergeometric distribution is calculated using the formula: P(x = k) = [(C(r, k) * C(N-r, n-k)) / C(N, n)]. For x = 0, k = 0, so P(x = 0) = [(C(6, 0) * C(4, 3)) / C(10, 3)] = [(1 * 4) / 120] = 0.033.

Question 8

Suppose a Poisson probability distribution with ? = 0.8 provides a good approximation of the distribution of a random variable x.. Find ? for x.

Accepted Answer

The mean of a Poisson distribution is equal to the variance. Therefore, the variance is also 0.8. The standard deviation is the square root of the variance, which is 0.8944.

Question 9

The conditions for both the hypergeometric and the binomial random variables require that the trials are independent.

Accepted Answer

The hypergeometric distribution does not require that the trials be independent, as it models situations without replacement, affecting the probabilities of subsequent trials. The binomial distribution assumes independent trials.

Question 10

Suppose a Poisson probability distribution with ? = 9.6 provides a good approximation of the distribution of a random variable x. Find ? for x.

Accepted Answer

The answer of Suppose a Poisson probability distribution with ?...

Question 11

The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 7.5. Interpret the value of the mean.

Accepted Answer

The mean of a Poisson distribution represents the average number of events (tickets) expected to occur in a fixed interval (per day). Here, it indicates that, on average, 7.5 tickets are written per day.

Question 12

The number of homeruns hit during a major league baseball game follows a Poisson distribution with λ = 3.2. Find the mean and standard deviation for this distribution.

Accepted Answer

For a Poisson distribution, the mean is λ, and the standard deviation is the square root of λ. Thus, mean = 3.2 and standard deviation = √3.2 ≈ 1.79.

Question 13

Given that x is a hypergeometric random variable, compute p(x) for N = 8, n = 5, r = 3, and x = 2.

Accepted Answer

p(x) = (C(r, x) * C(N-r, n-x)) / C(N, n)= (C(3, 2) * C(5, 3)) / C(8, 5)= (3 * 10) / 56= 30 / 56= .536

Question 14

The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 12 elements containing 7 successes. The numbers 0, 1, 2, 3, 4, 5, 6, and 7 are all possible values of x.

Accepted Answer

The hypergeometric random variable x in this scenario can only take on values from 0 to 5, because only 5 elements are being drawn from the set.

Question 15

As part of a promotion, both you and your roommate are given free cellular phones from a batch of 13 phones. Unknown to you, four of the phones are faulty and do not work. Find the probability that one of the two phones is faulty.

Accepted Answer

The probability that one of the two phones is faulty can be calculated by considering the combinations of selecting one faulty and one non-faulty phone. There are 4 faulty phones and 9 non-faulty phones. The probability is calculated as (4/13) * (9/12) + (9/13) * (4/12) = 0.462.

Question 16

Suppose x is a random variable for which a Poisson probability distribution with λ = 3 provides a good approximation. a. Graph p(x) for x = 0, 1, 2, 3, 4, 5, 6. b. Find μ and σ for x. c. What is the probability that x will fall in the interval μ ± σ?

Accepted Answer

The answer of Suppose x is a random variable for...

Question 17

Suppose the candidate pool for two appointed positions includes 6 women and 9 men. All candidates were told that the positions were randomly filled. Find the probability that two men are selected to fill the appointed positions.

Accepted Answer

The probability of selecting a man for the first position is 9/15. The probability of selecting a man for the second position is 8/14. The probability of selecting two men is (9/15) * (8/14) = .343.

Question 18

The hypergeometric random variable x counts the number of successes in the draw of n elements from a set of N elements containing r successes.

Accepted Answer

This is the definition of a hypergeometric random variable.

Question 19

A bank offers online banking to its customers free of charge. While online, customers can also sign up for additional services that the bank offers. Let x be the number of customers who sign up for additional services online each day. Suppose the distribution of x is approximated well by a Poisson distribution with mean λ = 42.3. Find E(x) and interpret its value.

Accepted Answer

The answer of A bank offers online banking to its...

Question 20

The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 10 elements containing 2 successes. List the possible values of x.

Accepted Answer