Prob theory Final

Sample space definition

the set of all possible outcomes of an experiment

permutation vs combination

permutation: order matters. n!/(n-k)!

combination: order doesn’t matter. n!/(k!(n-k)!). (n k)

What is stars and bars and how do you use it?

Number of ways to put n identical objects into k distinct bins
Number of ways to put n identical objects into k distinct bins when number in each bin can be 0. (n+k-1 k-1)

Number of ways to put n identical objects into k distinct bins when number in each bin must be >=1. (n-1 k-1)

Formula for conditional probability P(A|B)

P(A|B)=P(A and B)/P(B)

Baye’s rule

P(A|B)*P(B)/P(A)

Law of total probability

How to prove independence

P(A |B)=P(A)

P(A and B)= P(A)P(B)

When to use

binomial

hypergeometric

geometric

negative binomial

poisson

uniform

exponential

binomial: success in n independent trials. p=probability of success. Bin(n,p)

hypergeometric: success in n trials WITHOUT REPLACEMENT.

geometric: number of trials until 1st success; each trial having a constant success probability of p and failure probability of q (= 1-p)

negative binomial: number of FAILURES until rth success

poisson: # of events in a predefined fixed interval X~Pois(lambda)

uniform: all outcomes in [a,b] equally likely

exponential: time between events in a poisson process

Explain memoryless property. WHich distributions does it apply to

Probability of a future waiting time is independent of time already elapsed

exponential (continuous) and geometric (discrete)

Describe poisson process. What the poisson and exponential distributions represent

Stochastic model for counting random, independent events at a constant average rate (lambda) over time or space.

Number of arrivals in time t is Pois(lambda*t)

Time between arrivals are iid Exp(lambda)

How to prove independence of joint distributions

P(X=x,Y=y)=P(X=x)P(Y=y)

How to find marginal distribution of joint distributions

P(X=x)=P(X=x,Y=all values of y)

P(Y=y)=P(X=all values of x, Y=y)