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# Probability Revisited - PowerPoint PPT Presentation

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

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### Probability Revisited

Austin Cole

• Expectation & Variance

• Distributions

• Bernoulli

• Binomial

• Geometric

• Negative Binomial

• Hypergeometric

• Poisson

• 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)

• 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 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 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 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 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 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 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 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 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 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 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

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 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 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 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 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 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?