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

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Presentation Transcript
outline
Outline
  • Expectation & Variance
  • Distributions
    • Bernoulli
    • Binomial
    • Geometric
    • Negative Binomial
    • Hypergeometric
    • Poisson
probability basics
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
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
slide5

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
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
Bernoulli Distribution
  • K=1 signifies ‘success’, K=0 represents failure
  • Whether a coin comes up heads
  • What is f(x)?
bernoulli distribution1
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
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 distribution1
Binomial Distribution
  • E[X]=np
  • V[X]=np(1-p)
  • Example: Defective eggs
slide12

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
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
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
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
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
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

example1
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
Poisson Distribution
  • Often used for large n and small p
  • E[X]=λ
  • V[X]= λ
  • f(X=k)=
a closer look at poisson
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
slide24

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