Discrete Probability Distributions

Question 1

Uniform

Answer 1

all events are equally likely; constant probability

Question 2

Bernoulli

Answer 2

> 2 outcomes, success or failure
P(X=1) = p
P(X=0) = 1 - p

Question 3

Binomial assumptions

Answer 3

> All trials are independent
All trials have the same probability of success
Two outcomes on each trial

Question 4

Binomial definition

Answer 4

> X=number of successes in n random trials

Question 5

Binomial parameters

Answer 5

X ~ (n,p)
where n = number of trials
p = probability of success

Question 6

0!

Answer 6

1

Question 7

Geometric Definition

Answer 7

X = number of trials until first success is observed.

Question 8

Geometric assumptions

Answer 8

> Infinite number of possible trials
All trials are independent
All trials have same probability of success (p)

Question 9

Geometric parameters

Answer 9

X ~ Geometric(p)
where p = probability of success

Question 10

Negative Binomial definition

Answer 10

X = number of trials until rth success

Question 11

Negative Binomial Assumptions

Answer 11

> All trials are independent
All trials have same probability of success (p)
Requires a sequence of r-1 fails in x-1 trials, followed by success on the xth trial.

Question 12

Negative binomial parameters

Answer 12

X ~ Negative Binomial (p,r)
where p = probability of success
r = number of successes
x = total number of trials

Question 13

Poisson distribution definition

Answer 13

X = number of events over a fixed interval of time/space

Question 14

Poisson assumptions

Answer 14

> Events occur randomly at constant rate λ
Events are independent
Events occur uniformly

Question 15

Poisson Parameters

Answer 15

X ~ Poisson (λ)
where λ = average no. of events per unit interval

Question 16

Poisson approximation to Binomial

Answer 16

> when 𝑝 < 0.1 and 𝑛 > 50
the product 𝜆 = 𝑛𝑝 is constant
then the binomial(n, p) probabilities will be close to the Poisson(𝜆) probabilities
with 𝜆 = 𝑛𝑝
the Poisson can be used as an approximation.