Chapter 3

Question 1

What is a discrete random variable?

Answer 1

A discrete random variable is a function that assigns a real number to each outcome in the sample space of a random experiment.

Question 2

What does the sample space represent in a discrete random variable?

Answer 2

The sample space will be discrete.

Question 3

Provide an example of a sample space.

Answer 3

S = {a, b, c}
S = {yes, no}
S = {low, mid, high}

Question 4

In the context of a random experiment, what does the random variable represent?

Answer 4

The random variable represents the numerical outcome of a trial.

Question 5

What is the probability that a camera passes the test if the probability is given as 0.8?

Answer 5

0.8

Question 6

What is the probability that a camera fails the test if the probability of passing is 0.8?

Answer 6

0.2

Question 7

What is the random variable X in the example with cameras?

Answer 7

X denotes the number of cameras that pass the test.

Question 8

What is a probability distribution?

Answer 8

A probability distribution describes the probabilities associated with the possible values of a random variable.

Question 9

What does a probability mass function specify for a discrete random variable?

Answer 9

It specifies the list of possible values along with the probability of each.

Question 10

Define cumulative distribution function.

Answer 10

The cumulative distribution function is the probability that a random variable will be found at a value less than or equal to a certain number.

Question 11

What are the properties of a cumulative distribution function for a discrete random variable?

Answer 11

• Non-decreasing
• Limiting values: 0 as x approaches negative infinity and 1 as x approaches positive infinity
• Right-continuous

Question 12

What does the mean of a discrete random variable represent?

Answer 12

The mean is a measure of center or middle of the probability distribution.

Question 13

How is variance defined for a discrete random variable?

Answer 13

Variance is a measure of the dispersion or variability in the distribution.

Question 14

What is the expected value of a function of a discrete random variable?

Answer 14

It is calculated using the probability mass function.

Question 15

What is a discrete uniform distribution?

Answer 15

A discrete uniform distribution is where all outcomes are equally likely.

Question 16

Provide an example of a discrete uniform distribution.

Answer 16

The sample space for rolling a dice: S = {1, 2, 3, 4, 5, 6}.

Question 17

What are the requirements for a binomial distribution?

Answer 17

• The trials are independent
• Each trial results in only two possible outcomes
• The probability of success remains constant

Question 18

How is the probability mass function for a binomial random variable defined?

Answer 18

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

Question 19

What does the binomial coefficient represent?

Answer 19

It represents the number of ways to choose k successes in n trials.

Question 20

What is the probability of exactly 2 successes in 18 trials with a success probability of 0.1?

Answer 20

Use the binomial distribution formula with n = 18, k = 2, and p = 0.1.

Question 21

True or False: The expected value of a discrete random variable can be calculated using the weighted average of possible values.

Answer 21

True

Question 22

Fill in the blank: The mean of a discrete random variable is a _______.

Answer 22

[measure of center]

Question 23

Fill in the blank: The variance of a discrete random variable is a measure of _______.

Answer 23

[dispersion]

Question 24

What function is used in Microsoft Excel® to calculate the binomial distribution?

Answer 24

BINOMDIST

This function calculates the probability of a given number of successes in a specified number of trials.

Question 25

In a binomial distribution, what does the mean represent?

Answer 25

The mean is calculated as n * p ## Footnote Here, n is the number of trials and p is the probability of success.

Question 26

What is the formula for the variance of a binomial random variable?

Answer 26

Variance = n * p * (1-p) ## Footnote This formula expresses the variability of the binomial random variable.

Question 27

True or False: The trials in a hypergeometric distribution are independent.

Answer 27

False ## Footnote In a hypergeometric distribution, the trials are not independent as they are conducted without replacement.

Question 28

What is the probability that both parts conform when selecting from a batch of 850 parts with 50 nonconforming parts?

Answer 28

P(Both conform) ## Footnote This requires calculating the probabilities based on the hypergeometric distribution.

Question 29

Fill in the blank: The hypergeometric distribution is used when samples are selected from a _______ population without replacement.

Answer 29

finite ## Footnote This characteristic differentiates it from the binomial distribution.

Question 30

What are the two possible outcomes in a hypergeometric distribution?

Answer 30

* Success * Failure ## Footnote Each trial can result in one of these two outcomes.

Question 31

What is the probability of selecting all parts from the local supplier if 4 parts are chosen from a total of 300 parts?

Answer 31

P(All from local supplier) ## Footnote This probability is determined using the hypergeometric distribution.

Question 32

What are the important terms and concepts related to probability distributions mentioned in Chapter 3?

Answer 32

* Probability Distributions * Probability Mass Function * Cumulative Distribution Functions * Mean and Variance of a Random Variable * Discrete Uniform * Binomial * Hypergeometric ## Footnote These concepts form the foundation for understanding probability distributions.

Question 33

What does the cumulative distribution function (CDF) calculate in the context of binomial distribution?

Answer 33

The probability of obtaining a value less than or equal to a certain number ## Footnote The CDF is useful for determining probabilities across a range of outcomes.