What is a random variable?
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.
What is a discrete random variable?
Random variables that can assume a countable number of values are called discrete.
What is a continuous random variable?
Random variables that can assume values corresponding to any of the points contained in an interval are called continuous.
What is the probability distribution of a discrete random variable?
A graph, table, or formula that specifies that probability associated with each possible value that the random variable can assume.
What are the requirements for the probability distribution of a discrete random variable x?
- p(x) > 0 for all values of x.
- Σp(x) = 1
where the summation of p(x) is over all possible values of x.*
What is the mean, or expected value, of a discrete random variable x?
µ = E(x) = Σxp(x)
What is the variance of a random variable x?
σ^2 = E[(x-µ)^2] = Σ (x-µ)^2p(x) = Σx^2p(x) - µ^2
What are the Probability Rules for a Discrete Random Variable?
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
What are the characteristics of a Binomial Random Variable?
- The experiment consists of n identical trails.
- 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).
- 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.
- The trials are independent.
- The binomial random variable x is the number of S’s in n trials.
What is the Binomial Probability Distribution?
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)!
What are the Mean, Variance, and Standard Deviation for a Binomial Random Variable?
Mean: µ = np
Variance: σ^2 =npq
Standard Deviation: σ = √npq
What are the Characteristics of a Poisson Random Variable?
- 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).
- The probability that an event occurs in a given unit of time, area, or volume is the same for all the units.
- The number of events that occur in one unit of time, area, or volume is independent of the number that occur in other units.
- The mean (or expected) number of events in each unit is denoted by the Greek letter lambda (λ).
What are typical examples of poisson random variables?
- 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
What is the Probability Distribution, Mean, and Variance for a Poisson Random Variable?
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 . . .
What are the Characteristics of a Hypergeometric Random Variable?
- 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).
- The hypergeometric random variable x is the number of S’s in the draw of n elements.
What are examples of Hypergeometric Random Variables?
- 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.
What are Probability Distribution, Mean, and Variance of the Hypergeometric Random Variable?
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
Guide to Selecting a Discrete Random Variable Probability Distribution
Binomial - x = # of S’s in n trials
- n identical trials
- 2 outcomes: S,F
- P(S) & P(F) same across trials
- Trials independent
Poisson - x = # times a rare event (S) occurs in a unit
- P(S) remains constant across units
- Unit x-values are independent
Hypergeometric - x=# of S’s in n trials
- n elements drawn without replacement from N elements
- 2 outcomes: S,F.
