discrete probability distributions. discrete vs. continuous discrete ▫a random variable (rv) that...

Discrete Probability Distributions

Discrete vs. Continuous

•Discrete▫A random variable (RV) that can take only

certain values along an interval: Cars passing by a point Results of coin toss Students taking a class

•Continuous▫An RV that can take on any value at any

point along an interval: Temperature, time, distance, money etc.

Frequency Distribution

•Number of times an observation occurs in a given population.

What is a variable?

•A symbol (A, B, x, y, etc.) that can take on any of a specific set of values▫X=number of heads▫Y= temperature

•Random variable▫The outcome of a statistical experiment

Random variable notation

•Capital letter represents the RV▫X=total number of heads in 4 tosses▫P(X) represents the probability of X

•Lower-case letter represents one of the values of the RV▫P(X=x) is the probability the RV will

assume a specific value▫P(X=2) is the probability that we will have

exactly 2 heads in the 4 tosses

Probability Distribution

•Relative frequency distribution that should, theoretically, occur for observations from a given population.

Outcome #heads Probability

HH 2 .25

HT 1 .25

TH 1 .25

TT 0 .25

X P(X)

0 0.25

1 0.50

2 0.25

Cumulative probability distribution•Probability the value of a RV falls within a

specified range.•Coin toss: P(X≤1)

# Heads P(X=x) P(X≤x)

0 0.25 0.25

1 0.50 0.75

2 0.25 1.00

Characteristics of a Discrete Probability Distribution

•For any value of x

•The values of x are exhaustive, i.e. the distribution contains all the possible values

•The values of x are mutually exclusive; i.e., only one value can occur for an experiment

•The sum of the probabilities equals 1

Mean and standard deviation

•Mean of discrete distribution is called expected value

•Variance

•Standard Deviation

Practice

•Determine the Mean (µ) or Expected Value (E(x)) for the following data.

X 0 1 2

P(x) 0.6 0.3 0.1

Practice

•A music shop is holding a promotion in which the customer rolls a die and deducts a dollar from the price of a CD equal to the number that he rolls.

•If the owner pays $5.00 for each disk and prices them at $9.00, what will his expected profit be on each CD during this promotion?

Binomial Distributions

•There are 2 or more identical trials•In each trial, there can be only 2

outcomes (success or failure)•Trials are statistically independent

▫Outcome of one trial does not influence outcome of the next

•Probability of success remains the same from one trial to the next

Binomial Experiment?

•An article in a 1988 issue of The New England Journal of Medicine talked about a TB outbreak. ▫One person caught the disease in 1995▫232 workers sampled from a very large

population were given a TB test▫The number of workers testing positive is

the variable of interest•If we test all 232 workers for the disease,

is this a binomial experiment?

Binomial Experiment?

•Bill has to sell 3 cars to meet his monthly quota. He has 5 customers, but 3 of them are interested in the same car and will leave if that car is sold.

•He has a 30% chance of a sale with each customer.

•Is this a binomial experiment?


•Probability of exactly x successes in n trials:

•Where:▫π = probability of success for any trial▫n = number of trials▫x = number of successes▫(1-x) = number of failures


•Expected value

•Variance

Binomial Distributions in Excel• =binom.dist(number_s,trials,probability_s,cumul

ative)• Where:

▫ Number_s = number of successes▫ Trials▫ Probability_s = probability of success▫ Cumulative:

False, if we want the probability of x True, if we want the probability of all the variables up

to and including x

• Example, in the previous problem▫ P(X=5)▫ =binom.dist(5,5,.1,false)

Binomial Experiment

•We’re going to select 5 households at random in a city where the unemployment rate is 10% to see if the head of the household is unemployed. What is the probability that all 5 are employed?

•Is this a binomial experiment? ▫Why or why not?

Acceptance to college

•The probability that a student is accepted to a prestigious college is 0.3. If 5 students apply, what is the probability that at most 2 are accepted?

These examples came from the StatTrek website: http://stattrek.com/Lesson1/Statistics-Intro.aspx?Tutorial=StatThese examples came from the StatTrek website: http://stattrek.com/Lesson1/Statistics-Intro.aspx?Tutorial=StatThese examples came from the StatTrek website: http://stattrek.com/Lesson1/Statistics-Intro.aspx?Tutorial=Stat


0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

# Accepted

Pro

babi

lity

Cumulative Probability Distribution

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

# Accepted

Pro

babi

lity

Coin flipping - again

•What is the probability of getting 45 or fewer heads in 100 tosses of a fair coin?


0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99

-0.00999999999999998

1.90819582357449E-17

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Cumulative Probability Distribution

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 990

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

The World Series

•What is the probability that the World Series will last 4 games?

•5 games? •6 games? •7 games?

▫Assume the teams are evenly matched.

Poisson Distribution

•Applies for events occurring over time, space, or distance

•Examples:▫Number of cars driving past a point▫Number of defects per foot in

manufactured pipe▫Number of knots in a section of wood panel▫Number of accidents per day at a job site


e is the base of the natural logarithm system and is equal to 2.71828

Any number raised to a negative exponent is the same as 1 divided by that number raised to its exponent. Example: 2-2 is the same as 1/22


•There were 438 children born in a small town last year.

•What is the probability that, on any given day, no children were born?

Poisson Distribution in Excel

•=poisson.dist(x,mean,cumulative)•Where:

▫X=the number we’re looking for▫Mean = lambda▫Cumulative

True = probability of all values up to and including x

False = probability of x•=poisson.dist(0,1.2,false)


0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Cumulative Poisson Distribution•Suppose the average number of lions seen

on a 1-day safari is 5. What is the probability that tourists will see fewer than four lions on the next 1-day safari?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cumulative Poisson Distribution

Hypergeometric Distribution

•Sampling without replacement•Compare to binomial

▫There are 2 or more identical trials▫In each trial, there can be only 2 outcomes

(success or failure)▫Trials are statistically independent▫Probability of success remains the same

from one trial to the next▫The random variable is the number of

successes in n trials

Trials are not statistically independentProbability of success changes from one trial to the next


Where: N=size of the populationn=size of the samples=number of successes in the

populationx=number of successes in the

sample

Hypergeometric in Excel

•=hypgeom.dist(sample_s, number_sample, population_s, number_population, cumulative)

•Where:sample_s=number of successes in the

samplenumber_sample=size of the samplepopulation_s=number of successes in the populationnumber_population=size of the

populationcumulative=same as before

• =hypgeom.dist(2, 4, 6, 20, false)


•20 businesses filed tax returns•6 of the returns were filled out incorrectly•The IRS has randomly selected 4 of the 20

returns to audit•What is the probability that exactly 2 of

the 4 selected for audit will be filled out incorrectly?


0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Cumulative Hypergeometric

•Suppose we select 5 cards from an ordinary deck of playing cards. What is the probability of obtaining 2 or fewer hearts?

Cumulative Hypergeometric

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Summary

•Random variables▫Discrete v. continuous

•Probability distributions▫Cumulative distributions

•Expected values•Binomial•Poisson•Hypergeometric

discrete probability distributions. discrete vs. continuous discrete ▫a random variable (rv) that...

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