Chapter 5: Discrete Probability Distributions

Random Variables

A random variable is a numerical description of the outcome of an experiment.
Discrete Random Variable: May assume either a finite number of values or an infinite sequence of values.
- Example (JSL Appliances): x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4).
- Example (JSL Appliances): x = number of customers arriving in one day, where x can take on the values 0, 1, 2,…
Continuous Random Variable: May assume any numerical value in an interval or collection of intervals.

Random Variable x	Type
Family size x = Number of dependents reported on tax return	Discrete
Distance from home to store x = Distance in miles from home to store site	Continuous
Own dog or cat x = 1 if own no pet; = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)	Discrete

The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.
A discrete probability distribution can be described with a table, graph, or formula.
The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable.
The required conditions for a discrete probability function are:
- f(x) > 0
- \sum f(x) = 1
Example (JSL Appliances):
- A tabular representation of the probability distribution for TV sales was developed using past data.

The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula.
The discrete uniform probability function is f(x) = 1/n, where:
- n = the number of values the random variable may assume
- the values of the random variable are equally likely

The expected value, or mean, of a random variable is a measure of its central location.
The expected value is a weighted average of the values the random variable may assume. The weights are the probabilities.
The expected value does not have to be a value the random variable can assume.
E(x) = \mu = \sum xf(x)

The variance summarizes the variability in the values of a random variable.
The variance is a weighted average of the squared deviations of a random variable from its mean. The weights are the probabilities.
Var(x) = \sigma^2 = \sum (x - \mu)^2 f(x)
The standard deviation, \sigma, is defined as the positive square root of the variance.
Example (JSL Appliances):
- Expected number of TVs sold in a day: E(x) = 1.20
- Variance of daily sales: \sigma^2 = 1.660 TVs squared
- Standard deviation of daily sales: 1.2884 TVs

Four Properties of a Binomial Experiment:
1. The experiment consists of a sequence of n identical trials.
2. Two outcomes, success and failure, are possible on each trial.
3. The probability of a success, denoted by p, does not change from trial to trial (stationarity assumption).
4. The trials are independent.
Our interest is in the number of successes occurring in the n trials.
We let x denote the number of successes occurring in the n trials.
Binomial Probability Function:

f(x) = \frac{n!}{x!(n-x)!} p^x (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! = n(n – 1)(n – 2) ….. (2)(1)
Example: Evans Electronics
- Evans Electronics is concerned about a low retention rate for its employees. In recent years, management has seen a turnover of 10% of the hourly employees annually.
- Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year?
- Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year.
Binomial Probabilities and Cumulative Probabilities
- Statisticians have developed tables that give probabilities and cumulative probabilities for a binomial random variable.

A Poisson distributed random variable is often useful in estimating the number of occurrences over a specified interval of time or space
It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2,…).
Examples of a Poisson distributed random variable:
- The number of knotholes in 14 linear feet of pine board
- The number of vehicles arriving at a toll booth in one hour

Two Properties of a Poisson Experiment:
1. The probability of an occurrence is the same for any two intervals of equal length.
2. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.
Poisson Probability Function:

f(x) = \frac{m^x e^{-m}}{x!}

Example: Mercy Hospital
- Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings.
- What is the probability of 4 arrivals in 30 minutes on a weekend evening?

A property of the Poisson distribution is that the mean and variance are equal.
- \mu = \sigma^2

The hypergeometric distribution is closely related to the binomial distribution.
However, for the hypergeometric distribution:
- the trials are not independent, and
- the probability of success changes from trial to trial.
Hypergeometric Probability Function:

f(x) = \frac{\binom{r}{x} \binom{N-r}{n-x}}{\binom{N}{n}}

where:
- x = number of successes
- n = number of trials
- f(x) = probability of x successes in n trials
- N = number of elements in the population
- r = number of elements in the population labeled success
Hypergeometric Probability Function for 0 < x < r
- number of ways x successes can be selected from a total of r successes in the population
- number of ways n – x failures can be selected from a total of N – r failures in the population
- number of ways n elements can be selected from a population of size N
If these two conditions do not hold for a value of x, the corresponding f(x) equals 0.
However, only values of x where:
- 1. x < r and
- 2. n – x < N – r are valid.
The probability function f(x) on the previous slide is usually applicable for values of x = 0, 1, 2, … n.

Example: Neveready’s Batteries
- Bob Neveready has removed two dead batteries from a flashlight and inadvertently mingled them with the two good batteries he intended as replacements. The four batteries look identical.
- Bob now randomly selects two of the four batteries. What is the probability he selects the two good batteries?

\mu = E(x) = n(\frac{r}{N})

\sigma^2 = Var(x) = n(\frac{r}{N})(1 - \frac{r}{N})(\frac{N-n}{N-1})