Chapter 5 Discrete Probability Distributions

Section 5.1 Random Variables

Random Variable

-- numerical description of the outcome of an experiment

Discrete Random Variable

-- assumes either a finite number of values or an infinite sequence of values

Discrete Random Variable (finite # of values)

Example: # of voters in a given county election

Example: JSL Appliances

Let x = # of TV’s sold at the store in 1 day

Discrete Random Variable (infinite sequence of values)

Let x = # of customers arriving in one day

In some cases, you have to convert text answers to numeric through coding

For gender -> x = 1 for male x = 2 for female

Continuous Random Variables

-- may assume any numerical value in an interval or collection of intervals

Example: mileage, temperature, time, weight

Random Variable Summary Examples

Question/ExperimentRandom VariablesType

Family Sizex = # of dependents reported on tax return

Distance from homex = distance in miles from home to store

to store

Own a dog or catx = 1 if own no pet

x = 2 if own dog(s) only

x = 3 if own cat(s) only

x = 4 if own dog(s) and cat(s)

Section 5.2 Discrete Probability Distribution

-- can be described with a table, graph or equation

Probability Distribution

-- for a random variable, the probability distribution describes how probabilities are distributed over the values of the random variable

-- defined by a probability function, denoted as f(x), which provides the probability for each value of the random variable

For a discrete probability distribution,

f(x) 0

∑ f(x) = 1

Example

Graphically

Discrete Uniform Probability Distribution

-- simplest example of a discrete probability distribution, given by the formula:

f(x) = 1/n  values of random variable are equally likely

where n = # of values the random vaiable may assume

Example: Experiment is rolling a die

Section 5.3 Expected Value, Variance & Standard Deviation of Discrete Random Variable

Expected Value (Mean)

-- measure representing the central location of a random variable

-- known as the mean of a random variable

E(x) = μ = ∑x f(x)

Variance

-- summarizes the variability/dispersion in the values of a random variable

Var(x) = 2 = ∑ (x – μ)2 f(x)

Standard Deviation

Example: Expected Values of TV’s sold in one day

Section 5.4 Binomial Probability Distribution

4 Properties of a Binomial Experiment

1) Experiment consists of a sequence of n identical trials

2) Two outcomes, success and failure, are possible on each trial

3) Probability of a success, denoted by p does not change from trail to trial ( 1- p is the probability of a failure)

4) Trails are independent

Our interest is in the # of successes occurring in the n trials

Let x = # of successes occurring in n trials

Example: Evans Electronics

Evans is concerned about a low retention rate for employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. 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.

Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year?

Check Properties:

1) 3 identical trails

2) 2 outcomes possible (Leave/Stay)

3) Probability of success (leave) = .10

4) Trials are indept

-- All 4 are satisfied

Using Tables showing the probability of x successes in n trials (starting on p 978 in text)

Expected Values/Variance/Standard Deviation of Binomial Distribution

Expected Value

E(x) =  = np

Variance

Var(x) = 2 = np(1 - p)

Standard Deviation