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 \begin{align*}X\end{align*}
X is given by the following equation:\begin{align*}\mu=E(x)=\sum_{} xp(x)\end{align*}

Variance
The variance of a discrete random variable is given by the following formula:
\begin{align*}\sigma^2 = \sum_{} (x\mu)^2 P(x)\end{align*}

Standard Deviation
The square root of the variance, or, in other words, the square root of \begin{align*}\sigma^2\end{align*} , is the standard deviation of a discrete random variable:
\begin{align*}\sigma=\sqrt{\sigma^2}\end{align*}
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 \begin{align*}X\end{align*} 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: \begin{align*}\sigma=\end{align*} √(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 \begin{align*}(n, p, a)\end{align*} , where \begin{align*}n\end{align*} is the number of trials, \begin{align*}p\end{align*} is the probability of success for any particular trial, and \begin{align*}a\end{align*} 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 \begin{align*}(n, p, a)\end{align*}.
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:
\begin{align*}P & = \frac{n!}{n_1!n_2!n_3!\ldots n_k!} \times \left (p_1{^{n_1}} \times p_2{^{n_2}} \times p_3{^{n_3}} \ldots p_k{^{n_k}} \right )\end{align*}
where:
\begin{align*}n\end{align*} is the number of trials.
\begin{align*}p\end{align*} is the probability for each possible outcome.
\begin{align*}k\end{align*} 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 \begin{align*}X\end{align*} 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: \begin{align*}\sigma=\end{align*} √((1p)/p^{2})
Poisson Distribution
In a binomial distribution, if the number of trials, \begin{align*}n\end{align*} , gets larger and larger as the probability of success, \begin{align*}p\end{align*} , 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:
\begin{align*}p(x) &= \frac{\lambda^x e^{\lambda}}{x!} \quad x=0, 1, 2, 3, \ldots\\ \mu &= \lambda\\ \sigma^2 &= \lambda\end{align*}
where:
\begin{align*}\lambda=\end{align*} the mean number of events in the time, distance, volume or area
\begin{align*}e=\end{align*} the base of the natural logarithm
Student's T Distribution
When you use \begin{align*}s\end{align*} to estimate \begin{align*}\sigma\end{align*} , you must use \begin{align*}t\end{align*} instead of \begin{align*}z\end{align*} to complete the significance test for a mean.
In calculating the \begin{align*}t\end{align*} test statistic, we use the formula:
\begin{align*}t=\frac{\bar{x}\mu_0}{\frac{s}{\sqrt{n}}}\end{align*}
where:
\begin{align*}t\end{align*} is the test statistic and has \begin{align*}n1\end{align*} degrees of freedom.
\begin{align*}\bar{x}\end{align*} is the sample mean
\begin{align*}\mu_0\end{align*} is the population mean under the null hypothesis.
\begin{align*}s\end{align*} is the sample standard deviation
\begin{align*}n\end{align*} is the sample size
\begin{align*}\frac{s}{\sqrt{n}}\end{align*} is the estimated standard error
df = n  1