Probability Mass Function of the Binomial Distribution and its Support 

[image]

for [image]

Binomial Distribution:
Mean, Variance

Mean: np
Variance: np(1-p)

Binomial Distribution:
Median

Either

one of [image]

Binomial Distribution:
Mode

[image][image][image]

Binomial Distribution:

MGF

[image]

Negative Binomial Distribution:
Probability Mass Function and Support 

[image]

 [image]

where    r is the number of successes,

k is the number of failures.

Negative Binomial Distribution:
Mean and Variance

Mean:

Variance:

Negative Binomial Distribution:

When To Use

 
Use this distribution when you are looking for the probability of k failures and r successes in k+r Bernoulli Trials. 

Geometric Distribution:

When to use

Use this distribution when looking for the number X of Bernoulli Trials needed to get one success. 

Geometric Distribution:

Probability Mass Function

 [image]

Geometric Distribution:

Mean and Variance

Mean:  

 [image]

Variance:

Negative Binomial Distribution:
MGF

[image]

[image]

Hypergeometric Distribution:

When to Use

This distribution is used when

looking for the number of successes in a sequence of draws from a sample WITHOUT replacement. 

Hypergeometric Distribution:

Probability Mass Function

[image]

Hypergeometric Distribution:
Mean and Variance

Mean:
[image]

where m is the number of items (say, red balls) that you can possibly draw from. Assuming you want red balls.

Variance:

[image]

where N is the total population, n is the amount sampled.  

Poisson Distribution:

When to Use

This distribution is used when looking for the number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time of the last event. Also can be used for intervals in distance, area, volume. 

Poisson Distribution:

Probability Mass Function

 and

Support 

[image]

[image]

Poisson Distribution:

Mean  and Variance

Mean:

Poisson Distribution:
MGF

[image]

Discrete Uniform Distribution:

When to use

 This distribution is used when

all values of a finite set of possible values are equally probable.

Discrete Uniform Distribution:

Probability Mass Function

[image]

[image]

Discrete Uniform Distribution:

Mean, Median, Variance

Mean= Median =

Variance =

[image]

where n is the number of possible values. 

Continuous Uniform Distribution:

When To Use

This distribution is used when all intervals of the same length are equally probable. 

Continuous Uniform Distribution:

Probability Density Function 

[image]

Continuous Uniform Distribution:
Mean, Median and Variance

Mean = Median =

[image][image]

Variance:
[image]

Continuous Uniform Distribution:
MGF

[image]

Exponential Distribution:

When to Use

They describe the times between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate.

Exponential Distribution:

Probability Density Function,

Cumulative Distribution Function 

PDF: 

λe ^{− λx}

CDF:
1 − e ^{− λx}

Mean =

 Median =

Mode = 0

 Variance =

[image]

Exponential Distribution:

MGF

[image]

Chi-Square Distribution:

Probability Density 

[image]

 for

Chi-Square Distribution:

Mean, Median, , Variance

Mean: k
Median:  approx k - 2/3

Variance: 2k 

Beta Distribution:

Probability Density

The probability density function of the beta distribution is:

[image]

[image]

[image]

Beta Distribution:

Mean, Variance

Mean:

Variance:

Beta Distribution:

Applications

B(i, j) with integer values of i and j is the distribution of the i-th order statistic (the i-th smallest value) of a sample of i + j − 1 independent random variables uniformly distributed between 0 and 1. The cumulative probability from 0 to x is thus the probability that the i-th smallest value is less than x, in other words, it is the probability that at least i of the random variables are less than x, a probability given by summing over the binomial distribution with its p parameter set to x. This shows the intimate connection between the beta distribution and the binomial distribution.

Beta distributions are used extensively in Bayesian statistics, since beta distributions provide a family of conjugate prior distributions for binomial (including Bernoulli) and geometric distributions. The beta(0,0) distribution is an improper prior and sometimes used to represent ignorance of parameter values.

Pareto Distribution:

Application

Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. This idea is sometimes expressed more simply as the Pareto principle or the "80-20 rule" which says that 20% of the population owns 80% of the wealth^[1]. It can be seen from the probability density function (PDF) graph on the right, that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. This distribution is not limited to describing wealth or income distribution, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed:

• The sizes of human settlements (few cities, many hamlets/villages)
• File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)
• Clusters of Bose-Einstein condensate near absolute zero
• The values of oil reserves in oil fields (a few large fields, many small fields)
• The length distribution in jobs assigned supercomputers (a few large ones, many small ones)
• The standardized price returns on individual stocks
• Sizes of sand particles
• Sizes of meteorites
• Numbers of species per genus (There is subjectivity involved: The tendency to divide a genus into two or more increases with the number of species in it)
• Areas burnt in forest fires
• Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.

Pareto Distribution:

Pr(X>x), Probability Density Function

And Support

[image]

PDF: [image]

where k := a positive parameter(shape)

x_m := the minimum value of x.(scale)

 [image]

Mean:

 
Variance:

       [image] for k > 2

Log Normal Distribution:

What is it? /Applications

In probability and statistics, the log-normal distribution is the single-tailed probability distribution of any random variable whose logarithm is normally distributed. If X is a random variable with a normal distribution, then Y = exp(X) has a log-normal distribution; likewise, if Y is log-normally distributed, then log(Y) is normally distributed. (The base of the logarithmic function does not matter: if log_a(Y) is normally distributed, then so is log_b(Y), for any two positive numbers a, b ≠ 1.)

A variable might be modeled as log-normal if it can be thought of as the multiplicative product of many small independent factors. For example the long-term return rate on a stock investment can be considered to be the product of the daily return rates. In wireless communication, the attenuation caused by shadowing or slow fading from random objects is often assumed to be log-normally distributed.

Log Normal Distribution:

Probability Density

and Support 

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for

[image]

Log Normal Distribution:

Mean, Variance

Mean:

[image]

Variance:
[image]

Log Normal Distribution:

Median, Mode

 Median:

[image]

Mode:
[image]

Gamma Distribution:

Probability Density Function

and Support

[image]

 for parameters:
  beta > 0 rate (real)
[image] scale (real)

Gamma Distribution:

Summation and Scaling Properties

Summation

If X_i has a Γ(k_i, θ) distribution for i = 1, 2, ..., N, then

[image]

provided all X_i are independent.

The gamma distribution exhibits infinite divisibility.

[edit] Scaling

For any t > 0 it holds that tX is distributed Γ(k, tθ), demonstrating that θ is a scale parameter.

Gamma Distribution:

Mean, Variance

Mean = k/beta

(k is the scale param)
(beta is the rate param)

Variance = k/beta^2 

Weibull Distribution:

Why, Applications

In probability theory and statistics, the Weibull distribution^[2] (named after Waloddi Weibull) is a continuous probability distribution. It is often called the Rosin–Rammler distribution when used to describe the size distribution of particles. The distribution was introduced by P. Rosin and E. Rammler in 1933.^[3] The probability density function is:

[image]

for x > 0 and f(x; k, λ) = 0 for x ≤ 0, where k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential.

The Weibull distribution is often used in the field of life data analysis due to its flexibility—it can mimic the behavior of other statistical distributions such as the normal and the exponential. If the failure rate decreases over time, then k < 1. If the failure rate is constant over time, then k = 1. If the failure rate increases over time, then k > 1.

An understanding of the failure rate may provide insight as to what is causing the failures:

• A decreasing failure rate would suggest "infant mortality". That is, defective items fail early and the failure rate decreases over time as they fall out of the population.

• A constant failure rate suggests that items are failing from random events.

• An increasing failure rate suggests "wear out" - parts are more likely to fail as time goes on.

When k = 1, the Weibull distribution reduces to the exponential distribution. When k = 3.4, the Weibull distribution appears similar to the normal distribution.

Weibull Distribution:

Mean, Variance

Mean:

[image]

Variance:
[image]

Normal Distribution:

Parameters,  Support, Density

Normal Distribution:

Mean, Median, Mode, Variance