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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 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

- 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 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

- 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 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?

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