Probability Revisited

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# Probability Revisited - PowerPoint PPT Presentation

Probability Revisited. Austin Cole. Outline. Expectation &amp; Variance Distributions Bernoulli Binomial Geometric Negative Binomial Hypergeometric Poisson. Probability Basics.

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### Probability Revisited

Austin Cole

Outline
• Expectation & Variance
• Distributions
• Bernoulli
• Binomial
• Geometric
• Negative Binomial
• Hypergeometric
• Poisson
Probability Basics
• Probability Mass Function (PMF): function that gives the probability that a discrete random variable is equal to some value, f(x)=P[X = x]
• Cumulative Distribution Function (CDF): a function F(x)=P[X ≤ x]
• For continuous r.v., f(x)=F´(x)
Expectation
• E[X]: What you expect the average for X to be in the long run
• Also known as weighted average, population mean or μ

An urn contains 3 red balls and 4 blue balls. Balls are drawn at random without replacement. Let the random variable X be the trial # when the 1st red ball is drawn. Find E[X]

Variance
• σ2=Var(X)=E[X2] – (E[X]) 2
• The square of the standard deviation σ of X
• How to calculate E[X2]?
• E[X2]=Σx2f(x) or ʃx2f(x)dx
Bernoulli Distribution
• K=1 signifies ‘success’, K=0 represents failure
• Whether a coin comes up heads
• What is f(x)?
Bernoulli Distribution
• E[X]=p
• V[X]=p(1-p)
• Special case of p=1/2
• μ=1/2
• V[X]=1/4 *largest possible variance for Bernoulli r.v.
• The PMF has the widest peak about the mean of any r.v.
Binomial Distribution
• Consists of n identical trials
• There are two possible outcomes
• Trials are mutually independent
• Probability of each success on each trial is the same
• f(X=k)=
Binomial Distribution
• E[X]=np
• V[X]=np(1-p)
• Example: Defective eggs

A dozen eggs contains 3 defectives. If a sample of 5 is taken with replacement, find the probability that exactly 2 of the eggs sampled are defective. Also, find the probability that 2 or fewer are defective.

• n=5; p=1/4
• f(x)=( )(1/4) x(3/4) 5-x
• Exercise 1

5

x

Geometric Distribution
• Probability that the first success comes on the kth trial
• f(X=k)=(1-p) k-1p
• E[X]=(1-p)/p
• V[X]=(1-p)/p 2
• Memoryless
Example
• Suppose the probability of an engine malfunction for any one-hour period is p=.02. Find the probability that a given engine will survive 2 hours.
• P[survive 2 hrs]=1-P[x<2]

=1-(.98)1-1(.02)-(.98) 2-1(.02)

=.9604

Negative Binomial Distribution
• Probability of having k successes and r failures
• E[X]=k(1-p)/p
• V[X]= k(1-p)/p 2
• f(X=k)=

k

Exercise 2
• A geological study indicates that an exploratory oil well drilled in a particular region should strike oil with probability p=.2. Find the probability that the 3rd oil strike comes on the 5th well drilled.
Hypergeometric Distribution
• Probability of sampling involving N items without replacement
• f(X=k)=
• m successes, N-m failures
• E[X]=nm/N
• V[X]=n*(--)*(1- --)*(----)

m

N

m

N

N-n

N-1

Example
• A biologist uses a “catch & release” program to estimate the population size of a particular animal in a region. During the catch phase, 20 animals are tagged. Months later, 30 animals are captured, and 7 have tags.
Poisson Distribution
• Often used for large n and small p
• E[X]=λ
• V[X]= λ
• f(X=k)=
A closer look at Poisson
• Suppose we want to find the probability distribution of the number of accidents at an intersection during the time period of one week
• Divide the week into subintervals so:
• P[no accident in subinterval]=1-p
• P[1 accident in subinterval]=p
• P[2+ accidents in subinterval]=0

Occurrence of accidents can be assumed to be independent from interval to interval (X~Bin(n,p))

• X=total # of subintervals w/ an accident
• Let p=λ/n
• ( )(--) 2 (1- --) n-x = (e-λ)*(λx)/x!

n

x

λ

n

λ

n

Poisson Example
• A rare disease affects .2% of the population. Find the probability that city A of 500,000 has 1,040 or fewer people infected.
• P(X≤1040)=Σ( )(.002 x).998 500000-x
• P(X≤1040)=Σ ------------

1040

X=0

500000

x

1040

X=0

1000x e-1000

x!

≈.8995

Discussion
• Are there any other uses that you see for probability?
• Have you used basic knowledge for probability in certain situations?