A distribution is a description of the possible values of a random variable and the possible occurrences of these values.
A probability distribution is a graph or table that specifies the probablity associated with each possible value that the random variable can assume.

Mean Value or Expected Value
The mean value, or expected value , of a discrete random variable is given by the following equation:

Variance
The variance of a discrete random variable is given by the following formula:

Standard Deviation
The square root of the variance, or, in other words, the square root of , is the standard deviation of a discrete random variable:
A sampling distribution is the probability distribution of a given statistic based on a random sample.
Normal Distribution
A normal distribution is called a bell curve because its shape is comparable to a bell. It has this shape because the majority of the data is concentrated at the middle and slowly decreases symmetrically on either side. A normal distribution can be described by the mean and standard deviation of the data.
Here is an example of a normal distribution:
The line in the middle represents the mean, and the tick lines along the bottom represent the standard deviation.
Binomial Distribution
Characteristics of a Binomail Probability Distribution
 The experiment consists of a fixed number of independent trials
 Each trial results in one of two outcomes: success or failure
 The probability of success p is the same for each trial
 The binomial random variable is defined as the number of successes in n trials
The binomial distribution is found by calculating the binomial probabilities for k = 0, 1, 2,..., n. It can be represented in a histogram with k along the xaxis and P(k) along the yaxis.
When n * p > 10 and n(1p) > 10, the following statements are all true:
 the normal distribution will provide a good approximation of the binomial distribution
 Mean for the binomial distribution: μ = np
 Standard deviation for the binomial distribution: √(np(1p))
The binompdf function on the TI83/84 calculator can be used to solve problems involving the probability of a precise number of successes out of a certain number of trials. The syntax for the binompdf function is binompdf , where is the number of trials, is the probability of success for any particular trial, and is the number of successes.
The binomcdf function on the TI83/84 calculator can be used to solve problems involving the probability of less than or equal to a number of successes out of a certain number of trials. The syntax for the binomcdf function is binomcdf .
Multinomial Distribution
The multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.
This probability is given by:
where:
is the number of trials.
is the probability for each possible outcome.
is the number of possible outcomes.
Geometric Distribution
Characteristics of a Geometric Probability Distribution
 The experiment consists of a sequence of independent trials
 Each trial results in one of two outcomes: success or failure
 The probability of success p is the same for each trial
 The geometric random variable is defined as the number of trials until the first success is observed
The geometric distribution is found by calculating the geometric probabilities for k = 0, 1, 2, ..., ∞ . It can be represented in a histogram with k along the xaxis and P(k) along the yaxis. Note: As k increases, P(k) approaches 0, so as this happens you can stop your calculations of P(k) for the purpose of creating the distribution.
Mean for the binomial distribution: μ = 1/p
Standard deviation for the binomial distribution: √((1p)/p^{2})
Poisson Distribution
In a binomial distribution, if the number of trials, , gets larger and larger as the probability of success, , gets smaller and smaller, we obtain a Poisson distribution.
Characteristics of a Poisson distribution:
 The experiment consists of counting the number of events that will occur during a specific interval of time or in a specific distance, area, or volume.
 The probability that an event occurs in a given time, distance, area, or volume is the same.
 Each event is independent of all other events.
Poisson Random Variable:
The probability distribution, mean, and variance of a Poisson random variable are given as follows:
where:
the mean number of events in the time, distance, volume or area
the base of the natural logarithm
Student's T Distribution
When you use to estimate , you must use instead of to complete the significance test for a mean.
In calculating the test statistic, we use the formula:
where:
is the test statistic and has degrees of freedom.
is the sample mean
is the population mean under the null hypothesis.
is the sample standard deviation
is the sample size
is the estimated standard error
df = n  1