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Probability Revisited

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

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Probability Revisited

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  1. Probability Revisited Austin Cole

  2. Outline • Expectation & Variance • Distributions • Bernoulli • Binomial • Geometric • Negative Binomial • Hypergeometric • Poisson

  3. 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)

  4. 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

  5. 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]

  6. 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

  7. Bernoulli Distribution • K=1 signifies ‘success’, K=0 represents failure • Whether a coin comes up heads • What is f(x)?

  8. 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.

  9. 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)=

  10. Binomial Distribution • E[X]=np • V[X]=np(1-p) • Example: Defective eggs

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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.

  16. 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

  17. 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.

  18. Poisson Distribution • Often used for large n and small p • E[X]=λ • V[X]= λ • f(X=k)=

  19. PMF CDF

  20. 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

  21. 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

  22. 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

  23. 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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