Chapter 5 - Discrete Probability Distribution p.219

Question 1

Binomial experiment p.244

Answer 1

An experiment having four properties
1. The experiment consists of a sequence of n identical trials.
2. Two outcomes are possible on each trial. We refer to one outcome as a success and the other outcome as a failure.
3. The probability of a success, denoted by p, does not change from trial to trial. Consequently, the probability of a failure, denoted by 1 - p, does not change from trial to trial.
The trials are independent.

Question 2

Binomial probability distribution p.244

Answer 2

A probability distribution showing the probability of x successes in n trials of a binomial experiment.

Question 3

Binomial probability function p.248

Answer 3

The function used to compute binomial probabilities.

Question 4

Bivariate probability distribution p.234

Answer 4

A probability distribution involving two random variables. A discrete bivariate probability distribution provides a probability for each pair of values that may occur for the two random variables.

Question 5

Continuous random variable p.222

Answer 5

A random variable that may assume any numerical value in an interval or collection of intervals.

Question 6

Discrete random variable p.221

Answer 6

A random variable that may assume any numerical value in an internal or collection of intervals.

Question 7

Discrete uniform probability distribution p.226

Answer 7

A probability distribution for which each possible value of the random variable has the same probability.

Question 8

Empirical discrete distribution p.224

Answer 8

A discrete probability distribution showing the probability of x successes in n trials from a population with r successes and N - r failures.

Question 9

Expected value p.229

Answer 9

A measure of the central location of a random variable.

Question 10

Hypergeometric probability distribution p.258

Answer 10

• A probability distribution showing the probability of x successes in n trials from a population with r successes and N - r failures.
• Closely related to the binomial distribution. The two probability distributions differ in two key ways:
  1. With the hypergeometric distribution, the trials are not independent
  2. And the probability of success changes from trial to trial

Question 11

Hypergeometric probability function p.258

Answer 11

• The function used to compute hypergeometric probabilities.
• Used to compute the probability that in a random selection of n elements. Selected without replacement, we obtain x elements labeled success and n - x elements labeled failure.
• f(x), the probability of obtaining x successes in n trials.

Question 12

Poisson probability distribution p.254

Answer 12

A probability distribution showing the probability of x successes in n trials of a binomial experiment.

Question 13

Poisson probability function p.254

Answer 13

The function used to compute binomial probabilities.

Question 14

Probability distribution p.224

Answer 14

A description of how the probabilities are distributed over the values of the random variable.

Question 15

Probability function p.224

Answer 15

A function, denoted by f(x), that provides the probability that x assumes a particular value for a discrete random variable.

Question 16

Standard deviation p.230

Answer 16

The positive square root of the variance.

Question 17

Variance p.229

Answer 17

A measure of the variability, or dispersion, of a random variable.

Question 18

5.3 Discrete Uniform Probability Function

Answer 18

• f(x) = 1/n
• Where
  - N = the number of values the random variable may assume.

Question 19

5.4 Expected Value of a Discrete Random Variable

Answer 19

E(x) = u = Sum(x*f(x))

Question 20

5.5 Variance of a Discrete Random Variable

Answer 20

Var(x) = o2 = Sum((x - u)2 * f(x))

Question 21

5.6 Covariance of Random Variables x and y

Answer 21

The covariance and/or correlation coefficient are good measures of association between two random variables.

Covariance(x,y) = [Var(x + y) - Var(x) - Var(y)]/2

Question 22

5.7 Correlation between Random Variables x and y

Answer 22

To get a better sense of the strength of the relationship between two random variables we can compute the correlation coefficient.

Correlation(x,y) = Covariance(x,y)/(StDev(x) * StDev(y))

Question 23

5.8 Expected Value of a Linear Combination of Random Variables x and y

Answer 23

E(ax + by) = aE(x) + bE(y)

- Where ‘a’ represents the coefficient of x and ‘b’ represents the coefficient of y in the linear combination.

Question 24

5.9 Variance of a Linear Combination of Two Random Variables

Answer 24

When the covariance between two random variables is known, we can compute the variance of a linear combination of two variables.

Var(ax + by) = a2Var(x) + b2Var(y) + (2ab)Covariance(x,y)

Question 25

5.10 Number of Experimental Outcomes Providing Exactly x Successes in n Trials

Answer 25

(n . x) = n!/(x!(n - x)!)

Where, n! = n(n - 1)(n - 2) . . . (2)(1)
And, by definition, 0! = 1

Question 26

5.12 Binomial Probability Function

Answer 26

f(x) = (n . x)px(1 - p)(n - x)

Where

    - X = the number of successes
    - P = the probability of a success on one trial
    - N = the number of trials
    - f(x) = the probability of x successes in n trials
    - (n  .  x) = n!/(x!(n - x)!)

Question 27

5.16 Hypergeometric Probability Function

Answer 27

f(x) = (r . x)((n - r) . (N - x))/(N . n)

Where

    - x = the number of successes
    - n = the number of trials
    - f(x) = the probability of x successes in n trials
    - N = the number of elements in the population
    - r = the number of elements in the population labeled success

Question 28

5.17 Expected Value for the Hypergeometric Distribution

Answer 28

E(x) = u = n(r/N)

Question 29

5.18 Variance for the Hypergeometric Distribution

Answer 29

Var(x) = StandardDeviation^2 = o^2 = n(r/N)(1 - r/N)((N - n)/(N - 1))

Question 30

Random Variable

Answer 30

A random variable is a numerical description of the outcome of an experiment.

Question 31

Required conditions for a discrete probability function

Answer 31

f(x) >= 0

Sum(f(x)) = 1

Question 32

Properties of a Poisson Experiment

Answer 32

1. The probability of an occurrence is the same for any two intervals of equal length.
2. The occurrence or nonoccurrence in any internal is independent of the occurrence or nonoccurrence in any other interval.