Chapter 4

Question 1

What is a random variable?

Answer 1

A variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point.

Question 2

What is a discrete random variable?

Answer 2

Random variables that can assume a countable number of values are called discrete.

Question 3

What is a continuous random variable?

Answer 3

Random variables that can assume values corresponding to any of the points contained in an interval are called continuous.

Question 4

What is the probability distribution of a discrete random variable?

Answer 4

A graph, table, or formula that specifies that probability associated with each possible value that the random variable can assume.

Question 5

What are the requirements for the probability distribution of a discrete random variable x?

Answer 5

1. p(x) > 0 for all values of x.
2. Σp(x) = 1
   where the summation of p(x) is over all possible values of x.*

Question 6

What is the mean, or expected value, of a discrete random variable x?

Answer 6

µ = E(x) = Σxp(x)

Question 7

What is the variance of a random variable x?

Answer 7

σ^2 = E[(x-µ)^2] = Σ (x-µ)^2p(x) = Σx^2p(x) - µ^2

Question 8

What are the Probability Rules for a Discrete Random Variable?

Answer 8

Let x be a discrete random variable with probabilitydistribution p(x), mean µ, and stand deviation σ. Then, depending on the shape of p(x), the following probability statements can be made:

Chebyshev’s Rule: Applies to any probability distribution
Empirical Rule: Applies to probability distributions that are mound shaped and symmetric

P(µ - σ < x < µ + σ) - Chebyshev’s - >0, Empirical’s - about .68

P(µ - 2σ < x < µ + 2σ) - Chebyshev’s - >3/4, Empirical’s - about .95

P(µ - 3σ < x < µ + 3σ) - Chebyshev’s - >8/9, Empirical’s - about 1.00

Question 9

What are the characteristics of a Binomial Random Variable?

Answer 9

1. The experiment consists of n identical trails.
2. There are only two possible outcomes on each trial. We will denote one outcome by S (for Success) and the other by F (for Failure).
3. The probability of S remains the same from trial to trial. This probability is denoted by p, and the probability of F is denoted by q = 1 - p.
4. The trials are independent.
5. The binomial random variable x is the number of S’s in n trials.

Question 10

What is the Binomial Probability Distribution?

Answer 10

p(x) = (nvx)p^xq^(n-x) (x=0,1,2,…,n)

where 
p = Probability of a success on a single trial
q = 1 - p
n = Number of trials
x = Number of successes in n trials
n-x= Number of failures in n trials
(nvx) = n!/x!(n-x)!

Question 11

What are the Mean, Variance, and Standard Deviation for a Binomial Random Variable?

Answer 11

Mean: µ = np
Variance: σ^2 =npq
Standard Deviation: σ = √npq

Question 12

What are the Characteristics of a Poisson Random Variable?

Answer 12

1. The experiment consists of counting the number of times a certain event occurs during a given unit of time or in a given area or valume (or weight, distance, or any other unit of measurement).
2. The probability that an event occurs in a given unit of time, area, or volume is the same for all the units.
3. The number of events that occur in one unit of time, area, or volume is independent of the number that occur in other units.
4. The mean (or expected) number of events in each unit is denoted by the Greek letter lambda (λ).

Question 13

What are typical examples of poisson random variables?

Answer 13

• The number of traffic incidents per month at a busy intersection
• The number of noticeable surface defects (scratches, dents, etc.) found by quality inspectors on a new automobile
• The number of parts per million (ppm) of some toxin found in the water or air emissions from a manufacturing plant
• The number of diseased trees per acre of a certain woodland
• The number of death claims received per day by an insurance company
• The number of unscheduled admissions per day to a school

Question 14

What is the Probability Distribution, Mean, and Variance for a Poisson Random Variable?

Answer 14

p(x) = λ^xe^(-λ)
(x = 0,1,2,... )
µ = λ
σ^2 = λ

where
λ = Mean number of events during a given unit of time, area, volume, etc.
e = 2.71828 . . .

Question 15

What are the Characteristics of a Hypergeometric Random Variable?

Answer 15

1. The experiment consists of randomly drawing n elements without replacement from a set of N elements, r of which are S’s (for Success) and (N-r) of which are F’s (for Failure).
2. The hypergeometric random variable x is the number of S’s in the draw of n elements.

Question 16

What are examples of Hypergeometric Random Variables?

Answer 16

• define x as th number of women hired in a random selection of three applicants from a total of six men and four women.
• the number x of defective large-screen plasma televisions in a random selection of n=4 from a shipment of N=8 TVs.
• n=5 stocks are randomly selected from a list of N=15 stocks. Then the number x of the five companies selected that pay regular dividends to stockholders.

Question 17

What are Probability Distribution, Mean, and Variance of the Hypergeometric Random Variable?

Answer 17

p (x) = (rvx)(N-rvn-x)/(Nvn) [x = Maximum [0,n - (N -r)],…, Minimum (r,n)]

µ = nr/N 
σ^2 = r(N-r)n(N-n)/N^2(N-1)

where
N = Total number of elements
r = Number of S's in the N elements
n = Number of elements drawn
x = Number of S's drawn in the n elements

Question 18

Guide to Selecting a Discrete Random Variable Probability Distribution

Answer 18

Binomial - x = # of S’s in n trials

1. n identical trials
2. 2 outcomes: S,F
3. P(S) & P(F) same across trials
4. Trials independent

Poisson - x = # times a rare event (S) occurs in a unit

1. P(S) remains constant across units
2. Unit x-values are independent

Hypergeometric - x=# of S’s in n trials

1. n elements drawn without replacement from N elements
2. 2 outcomes: S,F.