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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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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



Variance
Variance drawn at random without replacement. Let the random variable X be the trial # when the 1

  • 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 drawn at random without replacement. Let the random variable X be the trial # when the 1

  • K=1 signifies ‘success’, K=0 represents failure

  • Whether a coin comes up heads

  • What is f(x)?


Bernoulli distribution1
Bernoulli Distribution drawn at random without replacement. Let the random variable X be the trial # when the 1

  • 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 drawn at random without replacement. Let the random variable X be the trial # when the 1

  • 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 drawn at random without replacement. Let the random variable X be the trial # when the 1

  • E[X]=np

  • V[X]=np(1-p)

  • Example: Defective eggs


5

x


Geometric distribution
Geometric Distribution 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.

  • 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 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.

  • 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 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.

  • 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 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.

  • 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 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.

  • 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 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.

  • 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 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.

  • Often used for large n and small p

  • E[X]=λ

  • V[X]= λ

  • f(X=k)=


Pmf cdf
PMF CDF 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.


A closer look at poisson
A closer look at Poisson 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.

  • 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


n

x

λ

n

λ

n


Poisson example
Poisson Example from interval to interval (X~Bin(n,p))

  • 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 from interval to interval (X~Bin(n,p))

  • Are there any other uses that you see for probability?

  • Have you used basic knowledge for probability in certain situations?


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