Chapter 6 - QMB2100

Question 1

What is a probability distribution?

Answer 1

A listing of all the outcomes of an experiment and the probability associated with each outcome.

Question 2

What are the characteristics of a probability distribution?

Answer 2

1. The probability of a particular outcome is between 0 and 1.
2. The outcomes are mutually exclusive.
3. The list of outcomes is exhaustive, so the sum of the probabilities of the outcomes is always 1.

Question 3

What is a random variable?

Answer 3

A variable measured or observed as the result of an experiment. By chance, the variable can have different values.

Question 4

What are the two types of random variables?

Answer 4

Discrete random variable and continuous random variable.

Question 5

What is a discrete random variable?

Answer 5

A random variable that can assume only certain clearly separated values.

Question 6

What is a continuous random variable?

Answer 6

A random variable that may assume an infinite number of values within a given range.

Question 7

What is the formula for the mean of a probability distribution?

Answer 7

μ = Σ[xP(x)]; where μ is the mean of a probability distribution, x is the value of a random variable; and P(x) is the probability of a particular value x.

Question 8

What is the formula for the variance of a probability distribution?

Answer 8

σ^2 = Σ[(x - μ)^2 * P(x)]; where where μ is the mean of a probability distribution, x is the value of a random variable, P(x) is the probability of a particular value x, and σ^2 is the variance.

Question 9

What is the formula for the standard deviation of a probability distribution?

Answer 9

σ = sqrt(σ^2); where σ^2 is the variance, and σ is the standard deviation.

Question 10

What are the 4 requirements of a binomial probability distribution?

Answer 10

1. There are only two possible outcomes on a particular experimental trial which are mutually exclusive.
2. The random variable is the number of successes for a fixed and known number of trials.
3. The probability of success is the same for each trial.
4. The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial.

Question 11

What is the formula for the binomial probability?

Answer 11

P(x) = nCx * π^x (1 - π)^n-x; where C denotes a combination, n is the number of trials, x is the random variable defined as the number of successes, π is the probability of success on each trial.

Question 12

What is the mean of a binomial distribution?

Answer 12

μ = nπ; where n is the number of trials and π is the probability of success on each trial.

Question 13

What is the variance of a binomial distribution?

Answer 13

σ^2 = nπ(1 - π); where n is the number of trials and π is the probability of success on each trial.

Question 14

What is the standard deviation of a binomial distribution?

Answer 14

σ = sqrt(σ^2); the square root of the variance.

Question 15

When can you use the hypergeometric distribution?

Answer 15

When the sample is selected from a finite population without replacement and if the size of the sample is more than 5% of the size of the population.

Question 16

What is the formula for the hypergeometric distribution?

Answer 16

P(X) = (sCx)*(N-sCn-x) / NCn ; where N is the population, n is the sample, S is the number of successes in the population, x is the number of successes in the sample.

Question 17

What are the 4 characteristics of the hypergeometric distribution?

Answer 17

1. An outcome on each trial of an experiment is classified into one of two mutually exclusive categories - true or false.
2. The random variable is the number of successes in a fixed number of trials.
3. The trials are not independent.
4. We assume that we sample from a finite population without replacement and n/N>0.05. So the probability of a success changes from each trial.

Question 18

What is the Poisson probability distribution?

Answer 18

A discrete probability distribution that describes the number of times some event occurs during a specified interval.

Question 19

What are the 2 assumptions for the Poisson probability distribution?

Answer 19

1. The probability is proportional to the length of the interval.
2. The intervals are independent.

Question 20

What are 3 characteristics of the Poisson probability distribution?

Answer 20

1. The random variable is the number of times some event occurs during a defined interval.
2. The probability of the event is proportional to the size of the interval.
3. The intervals do not overlap and are independent.

Question 21

What is the formula for Poisson probability distribution?

Answer 21

P(x) = (μ^x * e^-μ) / x!; where μ is the mean, e is the constant 2.71828, x is the number of occurrences.

Question 22

What is the mean for Poisson probability distribution?

Answer 22

μ = nπ

Question 23

What is the variance for Poisson probability distribution?

Answer 23

σ^2 = nπ

Question 24

What is the standard deviation for

Answer 24

Square root of σ^2.