1. Discrete Uniform Distribution
The discrete uniform distribution occurs when there are a finite number (m) of equally likely outcomes possible. The pmf of a uniform discrete random variable X is:
p(x) = 1 / m, where x=1,2,…,m
The mean and variance of a discrete uniform random variable X are:
µ = (m + 1) / 2
σ2 = (m2 - 1) / 12

2. Bernoulli Distribution
A random experiment with two possible outcomes that are mutually exclusive and exhaustive is called a Bernoulli Trial.
One outcome is arbitrarily labeled a “success” and the other a “failure”
p is the probability of a success
q = 1 - p is the probability of a failure
The Bernoulli random variable X assigns:
X(failure) = 0 and X(success) = 1

3. Bernoulli Distribution
The pmf for a Bernoulli random variable X is:
p(x) = px (1-p)1-x, where x=0,1
The mean and variance of a Bernoulli random variable X are:
µ = p
σ2 = pq

4. Binomial Distribution
A binomial experiment results from a sequence of n independent Bernoulli trials.
The probability of success (p) remains constant in a binomial experiment
The number of successes (X) is the random variable of interest in a binomial experiment
If Y1, Y2, …, Yn are independent Bernoulli random variables, then X=∑ Yi is a binomial random variable.

5. Binomial Distribution
The pmf for a Binomial random variable X is:
p(x) = nCx px (1-p)n-x, where x=0,1,…,n
The mean and variance of a binomial random variable X are:
µ = np
σ2 = npq

6. Hypergeometric Distribution
The hypergeometric distribution applies when sampling without replacement from two possible mutually exclusive and exhaustive outcomes.
Let X be the number of objects of type 1 drawn if n objects are drawn from N where there are M objects of type 1 and N-M objects of type 2. Then X is a hypergeometric random variable and the pmf of X is:

7. Hypergeometric Distribution
The mean and variance of a hypergeometric random variable X are:
µ = n · (M/N)
σ 2 = n · (M/N) · (1-M/N) · (N-n)/(N-1)
As M and N converge to infinity and (M/N) converges to p, the hypergeometric distribution with n samples converges to the binomial distribution with n trials and p=M/N.

8. Geometric Distribution
A geometric distribution occurs when sampling independent Bernoulli trials. If X is the number of Bernoulli trials until the first success is observed, then X is a geometric random variable with pmf:
p(x) = (1-p)x-1p, where x=1,2,3,…
The mean and variance of a geometric random variable X are:
µ = 1 / p
σ2 = q / p2

9. Geometric Distribution
Notice that for integer k,
P( X > k ) = qk
P( X ≤ k ) = 1 – qk
“Memoryless” or “No Memory” Property
If X is a geometric random variable, then
P( X > j + k | X > j ) = P( X > k )
This implies that in independent Bernoulli trails, there is no such thing as being “due” to observe a success.

10. Negative Binomial Distribution
A negative binomial distribution occurs when sampling independent Bernoulli trials. If X is the number of Bernoulli trials until the rth success is observed, then X is a negative binomial random variable with pmf:
p(x) = x-1Cr-1 pr (1-p)x-r, where x=r,r+1,r+2,…
The mean and variance of a negative binomial random variable X are:
µ = r (1/p)
σ2 = r (q / p2)

11. Poisson Distribution
The Poisson distribution describes the number of occurrences of an event in a given time or on a given interval.
Assumptions of a Poisson Process
The number of events occurring in non-overlapping intervals is independent.
The probability of 1 event occurring in a significantly short interval h is lh.
The probability of 2 events occurring in a significantly short interval h is essentially zero.

12. Poisson Distribution µ = s2 = l
If X is defined to be the number of occurrences of an event in a given continuous interval and is associated with a Poisson process with parameter l>0, then X has a Poisson distribution with pdf:
The mean and variance of a Poisson random variable X are:
µ = s2 = l

13. Poisson Distribution
If events of a Poisson process occur at a mean rate of l per unit, then the expected number of occurrences in an interval of length t is lt. Moreover, if Y is the number of occurrences in an interval of length t, it is Poisson with pdf:

14. Poisson Distribution
The Poisson distribution with parameter l=np is useful for approximating the binomial distribution with sample size n and probability of success p in cases with sufficiently large sample size (n>20 and p<0.05).
B(n,p) P(l=np) as n ∞, p ,
and np l