Rare Event Approach

Discrete Random Variables

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1) Define random variable and related key terms.

2) Discuss the presentation of probability distributions.

3) Calculate descriptive statistics for discrete RVs.

Expected value
Variance / standard deviation

4) Describe the characteristics of three special types of discrete RVs and the formulae used to calculate the associated descriptive statistics.

Binomial Distribution
Hypergeometric Distribution
Geometric Distribution

Rare Event Approach

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In order to use this approach to make inferences, we need to know:

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We can use this information to infer how likely we are to observe the particular outcome that results from our experiment.

Key Terms

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Experiment -an act or process of observations that leads to a single outcome that cannot be predicted with certainty

EX:

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.

Discrete RVs -

Continuous RVs -

Examples of Random Variables

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1)# of heads in two flips of a coin.

2)# of odd rolls in three rolls of a die.

3)Height of a person selected at random.

4)# of matches required to start a campfire

5)volume of milk produced by a cow per day

Describing a Discrete Random Variable

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1) List all possible values

2) List the P associated with each value

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Define x as a RV = the # of heads in two tosses of a coin.

x = 0P (0) = .25

x = 1P (1) = .50

x = 2P(2) = .25

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Define x as a RV = the # of heads in three tosses of a coin.

x = 0P (0) = .125

x = 1P (1) = .375

x = 2P(2) = .375

x = 3P(3) = .125Probability Distributions

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P(x)

.125

.375

.125

x

/ 0 / 1 / 2 / 3
P(x) / .125 / .375 / .375 / .125

Descriptive Statistics for Discrete RVs

The Expected Value - E(x)

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The mean, or expected value, of a discrete random variable is given by the following:

=

E(x) =

E(x)

Weighted Average: Wedding Example

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You are getting married. It is known that the friends and family of the Bride are cheap whereas the friends and family of the Groom are generous. If the 5 guests from the Bride’s side give gifts worth $20 each and the 5 guests from the Groom’s side give gifts worth $50 each, what is the average (expected) value per gift?

Calculating E(x): Coins in a fountain

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You are flipping coins into a fountain to determine whether s/he loves you, or s/he loves you not. If you flip three coins into the fountain, what is the expected number of coins that will come up heads (meaning that s/he loves you)?

x

/ 0 / 1 / 2 / 3
P(x) / .125 / .375 / .375 / .125

 = E(x) =  [x  p(x)]

=0(.125) + 1(.375) + 2(.375) + 3(.125)

=0 + .375 + .75 + .375

=1.5

Important point:

Calculating E(x): Taking advantage of Biff

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Poor Ol’ Biff. You need some extra cash so you make a wager with Biff based on flipping two coins. You tell Biff that you will give him a dollar if he gets two heads, but that he will owe you a dollar if he flips one head and one tail. Two tails will be considered a tie. How much money would you expect to win per flip? 10 flips? Estimate how many flips would be needed for you to walk away with $25?

P(x)

.25

.50

.25

Variance of a discrete random variable

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Variance of a discrete RV =

2 = E[(x-)2] =  (x-)2 p(x)

Standard Deviation of a discrete RV =

 = 2

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Biff Example

-1

P(x)

.25

.50

2 =  [(x-)2 p(x)]

= (-1-.25)2(.25) +(0-.25)2(.25) +(1-.25)2(.50)

= (.390625) + .015625 + .28125

2 = .6875

= .83

The Real Cookie Monster

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Every night after dinner, I eat a moderate to large quantity of Oreo cookies. Here is the probability distribution for the number of Oreos I eat on a given evening. Find E(x) and sd.

x / 2 / 3 / 4 / 5 / 6
p(x) / .10 / .15 / .35 / .10

E(x) =  =  [x * p(x)]

Var =  [(x - )2 * p(x)]

SD= 

Binomial Rule

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We use the binomial rule when we are conducting repeated trials of the same experiment and want to know how many times x (a particular outcome) will occur if we repeat the experiment n times.

How do you recognize an application for the binomial rule?

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1) The experiment consists of a fixed number of identical trials.

2) The trials must be independent.

3) There are a discrete number of possible outcomes all of which can be classified as either “successes” or “failures”.

4) The probability of a success cannot change from trial to trial.

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Notation:P(success)=

P(failure)=

Are these examples of Binomial RVs?

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1) The number of dresses a woman tries on before she purchases one.

2) The probability of witnessing traffic congestion at a given intersection every hour for a day.

3) The probability of correctly answering 4 of the 6 questions on a statistics quiz.

Binomial Formula

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n=# of trials

x=# of successes

p=P(success)

1-p=P(failure)

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Note: You only need three bits of information to calculate the probability of a binomial random variable.

Choose Shortcuts

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For big numbers:

Binomial RVs: Life is like a bag of marbles

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I have a bag of marbles containing 3 blue and 2 red. Let’s define x as a random variable that represents the number of ______selected in 2 draws (with replacement). Find the probability that x = 0? Find the probability that x = 1. Find the probability that x = 2.

X

/ 0 / 1 / 2
P(x)

1-p=

Wonderful Property of Binomial RVs

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Mean=E(x)==np

Var=2=npq

SD==npq

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Shaquille O'Neal is shooting free throws for charity. Your boss, decides to chip in a few bucks, but doesn't want to overdo it. He asks you to give him a rough estimate of how many shots Shaq Daddy is likely to make out of 10. Let's give poor, pathetic Shaq Fu the benefit of the doubt and assume that the Big Aristotle makes 60% of his free throws. Given that this is FantasyLand, we'll also assume independence. Calculate the mean and standard deviation for the number of free throws Kazaam makes.

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E(x) =  = np = 10.6=6

2= npq = 10.6.4=2.4

= npq = 2.4=1.55

Comparing the General and Shortcut Formulae

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Marble Example

X

/ 0 / 1 / 2
P(x)

General Formula

=[x(px)]

2= [(x-)2 p(x)]

=2=

Binomial Distribution Rule

= np==

2=npq==

=npq= =

Hypergeometric RVs

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We use the hypergeometric rule when we are interested in calculating the probability of a particular event occurring x times in n trials when sampling is done without replacement.

How do you recognize an application for the hypergeometric rule (comparison with binomial rule)?

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1) Experiment consists of a fixed number of identical trials.

2) There are a discrete number of possible outcomes all of which can be classified as either “successes” or “failures”.

3) Sampling occurs without replacement.

x) The probability of a success cannot change from trial to trial.

x) The trials must be independent.

Hypergeometric Formula

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NTotal number of elements

rnumber of successes in N elements

nnumber of elements drawn

xnumber of successes in n elements

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Note: you need four bits of information to calculate the probability of a hypergeometric random variable.

Sock Drawer

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The electricity has gone out in my house. I know that I have to teach later in the day so it’s somewhat important to me that my socks match. I remember from the day before that I have five socks strewn about my sock drawer. Two are brown, three are black. If I choose two socks at random (without replacement), what is the probability that both will be brown?

p(2) = rN-r

xn-x

p(2) = 25-2

22-2

p(2) = 11= 1 / 10 = .10

(5)(4) (3)(2)

(2)(3)(2)

Hypergeomatric RVs: Life is like a box of chocolates

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I have a box of chocolates with 3 chocolate truffles and 2 strawberry nougats. If you choose 2 candies at random (no squinching), what is the probability that you will get one of each?

Shortcut rules for Hypergeometric RVs

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Mean: = (n  r) / N

Variance:2 = r(N-r)n(N-n) / N2(N-1)

Standard Deviation: =  r(N-r)n(N-n) / N2(N-1)

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Sock Drawer

= (nr) / N=(22) / 5= .8

2=r(N-r)n(N-n) / N2(N-1)

2(5-2)2(5-2) / 52(5-1)

2(3)2(3) / 25(4)

36 / 100= .36

=.36= .6

Double Check using standard RV formula

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= E(x) =  x p(x)

=0(.3) + 1(.6) + 2 (.1) = .8

2= (x - )2 p(x)

(0 - .8)2(.3) + (1 - .8)2(.6) + (2 - .8)2(.1)

(.64)(.3) + (.04)(.6) + (1.44)(.1)

.192 + .024 + .144= .36

=.36= .6

Geometric Distribution

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Tells you the probability of obtaining the first success on the nth trial.

Formula:

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Same criteria as Binomial: distinguish the two by noting what the question asks!

Geometric Example: Viva Las Vegas

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You miss your 10:00 AM Spring Break flight to Vegas so you are forced to fly stand-by. There are flights leaving at 2:00, 4:00 and 6:00 later that day. The gate attendant tells you that there is a 40% chance that you will make it onto any one of the flights. What is the probability that you fly to Vegas on the 2:00 flight? The 4:00 flight? The 6:00 flight? What is the probability that you stay home for Spring Break?

P(1st)=p(1-p)x-1

P(2nd)=p(1-p)x-1

P(3rd)=p(1-p)x-1

P(miss them all)=