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

• 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 1 st red ball is drawn. Find E[X]

• σ 2 =Var(X)=E[X 2 ] – (E[X]) 2
• The square of the standard deviation σ of X
• How to calculate E[X 2 ]?
• E[X 2 ]= Σx 2 f(x) or ʃx 2 f(x)dx

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

• 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.

• 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)=

• 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

• Probability that the first success comes on the kth trial
• f(X=k)=(1-p) k-1 p
• E[X]=(1-p)/p
• V[X]=(1-p)/p 2
• Memoryless

• 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

• 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

• 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 3 rd oil strike comes on the 5 th well drilled.

• Probability of sampling involving N items without replacement
• f(X=k)=
• m successes, N-m failures
• E[X]=nm/N
• V[X]=n*(--)*(1- --)*(----)
N N N-1

• 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.

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

• 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
• ( )(--) x n
2 (1- --) n n-x = (e λ
)*( λ x )/x!

• A rare disease affects .2% of the population. Find the probability that city
A of 500,000 has 1,040 or fewer people infected.
1040
• P(X≤1040)=Σ( )(.002
x ).998
500000-x
X=0
500000 x
1040
• P(X≤1040)=Σ ------------
X=0
1000 x e -1000 x!
≈.8995

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