Statistical economics/Common Probability Density Functions

Probability distributions can be broken up into categories based on the continuity of the variables involved. This page breaks up these distributions by their use of discrete or continuous random variables.

General Notation

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Here are some notations used throughout this page:

  •   is the probability that the random variable   takes on the value  .
  •   is the expected value of the random variable  , and is equal to   for discrete random variables and   for continuous variables.
  • var  is the variance of the random variable  , and is a measure of the spread of the possible values of the variable. If   takes on a large range of values, then the variance is larger than if   only takes on a values relatively close to each other.

Probability Distributions of Discrete Random Variables

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Some outcomes only take on discrete values, commonly integers (but not always). These outcomes can be modeled by discrete random variables that can only take on those values, and which values appear with probabilities determined by the probability distribution that the random variable is characterized by. Examples of discrete outcomes are the number of cars passing a landmark each day, the number of customers buying more than $50 each day, or simply the number of heads that show up during a coin tossing contest.

Notes for Discrete Random Variables:

  •   is the probability that   takes on the value   (i.e.,  ). This is commonly referred to as the Probability Mass Function (PMF) for discrete random variables.

The Bernoulli Distribution

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This is possibly the simplest probability distribution, with only two possible outcomes: success or failure. A common example of a Bernoulli random variable is a coin toss. A Bernoulli distribution is described by the parameter  , which is the probability of success. Naturally,   is the probability of failure (if we were hoping for the coin to land on heads, this would be the probability that it lands on tails instead).

Notation

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If a random variable   follows a Bernoulli distribution, it can be denoted by  , where   is the probability of success or the favorable outcome. Conventionally, since this is a binary variable, we say that the result is a success when  , and a failure if  .

Distribution Details

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  •  
  •  
  •  
  • var 


The Binomial Distribution

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A Binomial random variable   can take on integer values from 1 to  , where   is the total number of identical trails, and   is the number of successful trials. This means that   can be understood as a sum of   Binomial random variables, each with probability   of success, where each Binomial random variable is the success or failure (1 or 0) of each individual trial. A common example of a Binomial random variable is the number of heads that result from flipping a coin   times, with the probability   of landing on heads.

Notation

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If a random variable   follows a Binomial distribution, it can be denoted by   or  , where   is the number of trials and   is the probability of success of the favorable outcome for each trial.

Distribution Details

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  •  
  •  
  •  
  • var 


The Geometric Distribution

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Notation

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If a random variable   follows a Geometric distribution, it can be denoted by  , where   is the probability of success of the favorable outcome.

Distribution Details

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  •  
  •  
  •  
  • var 


The Negative Binomial Distribution

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Notation

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If a random variable   follows a Negative Binomial distribution, it can be denoted by  , where   is the number of desired successes and   is the probability of success of the favorable outcome.

Distribution Details

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  •  
  •  
  •  
  • var 


The Poisson Distribution

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Notation

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If a random variable   follows a Poisson distribution, it can be denoted by  , where   is mean of the variable.

Distribution Details

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  •  
  •  
  •  
  • var 


The Hypergeometric Distribution

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Notation

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If a random variable   follows a Hypergeometric distribution, it can be denoted by  , where   is the total size of the population,   is the size of the sample taken, and   is the number of favorable objects in the population.

Distribution Details

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  •  
  •  
  •  
  • var 


The Uniform Distribution

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Notation

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If a random variable   follows a Uniform distribution, it can be denoted by  , where   is the total number of outcomes. A uniform distribution has equal probability over all possible outcomes, which is simply  .

Distribution Details

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  •  

If the set of outcomes contains only consecutive integers starting at 1, then:

  •  
  • var 


Probability Distributions of Continuous Random Variables

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Notes for Discrete Random Variables:

  •   is the probability density value that   takes on the value  . This is commonly referred to as the Probability Density Function (PDF) for continuous random variables. For continuous variables, since any range of real numbers has infinitely many numbers in it, the probability of   taking on any single number is 0. For continuous random variables, we instead ask about the probability that   will fall within some range of numbers. To get this probability, we define   as a density function that gives us probability when integrated. In other words, the probability that   is between the real numbers   and   is  .

The Uniform Distribution

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The Normal Distribution

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The Poisson Scatter Distribution

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The Exponential Distribution

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The Gamma Distribution

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The Chi-square Distribution

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The Beta Distribution

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The Distribution

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