Probability distributions

Question 1

normal distribution probability density function

Answer 1

Question 2

Poisson probability mass function

Answer 2

lambda = parameter, lambda>0

k is an integer, k is in [0, 1, 2, …]

Uses: Related to the number of occurrences of a poissson-type event over a fixed period of time

Question 3

Exponential family

(single parameter)

Answer 3

Question 4

Exponential family

(parameter vector)

Answer 4

Question 5

Binomial distribution probability mass function

Answer 5

n = # trials

p = success probability per trial, p is in [0,1]

Question 6

Expected value of a random variable X with a poisson distribution

Answer 6

E(X) = lambda

Recall Poisson:

Question 7

Variance of a random variable X with a poisson distribution

Answer 7

Var(X) = lambda

Recall the Poisson PMF is:

Question 8

Bernoulli distribution probability mass function

Answer 8

k can be 1 or 0 (coin flip)

On each trial, k is 1 with probability p and 0 wih probability 1-p

Expected value of a bernoulli distributed random variable

E(X) = p

Variance of a bernoulli distributed random variable

Var(X) = p(1-p)

Question 9

Exponential distribution

probability density function

Answer 9

Exponential distribution pdf

lambda = rate parameter, lambda>0

Properties: memoryless

E[X] = 1/lambda

Var[X] = 1/lambda^2

Uses: related to time between poisson-type events

Question 10

Exponential family pdf sketch

Answer 10

Question 11

Beta Distribution pdf

Answer 11

Only valid for 0<=x<=1

alpha and beta are called shape parameters

Properties:

E[X]=1/(1+beta/alpha)

Var[x] = complicated

Used for: random variables limited to finite intervals

Question 12

Beta distribution sketch

Question 13

Continuous uniform distribution pdf

(and expected value/ variance)

Answer 13

Properties:

E[X] = (a+b)/2

Var[X] = [(b-a)^2]/12

Question 14

Discrete uniform distribution pmf

Answer 14

Finite number of outcomes a to b inclusive

P(k) =