Geometric distribution

Definition of the geometric distribution

You're playing "Heads and Tails". You decide to play until the first "Head" turns up, and then stop the game. How many times do you have to toss the coin ?

After the first game, you decide to play a second game. You expect the number of tosses needed to "win" this second game to be different from the number of tosses you needed to win the first one. Therefore, L, the number of tosses needed to finally obtain a "Head" is a random variable. By definition, the probability distribution function of this r.v. is the geometric distribution.

Clearly, this probability distribution function depends on the bias of the coin. Let p be the probability for a "Head" (and therefore q = 1 - p the probability for a "Tail"). For a fair coin, p = 0.5. But suppose p = 0.9, you would expect, on the average, the games to be much shorter than for a fair coin.

You'll find here an interactive animation illustrating the Geometric Distribution  .

Basic properties of the geometric distribution

The basic properties of the geometric distribution are :

Probability mass function P{L = k}

 The probability for the first "Head" to turn up at toss #k is :
 

Cumulative distribution function F(n)

By definition, F(n) is the probability that it takes at most n tosses for the first "Head" to turn up.
 

Mean

Variance

Moment generating function of the geometric distribution

Memoryless property

The concept of "memoryless process" is explained here (within the context of the exponential distribution). This property is expressed by :

For the Geometric Distribution, this translates into :

• Suppose that after s tosses, no "Head" has turned up yet. Then the probability for no "Head" to turn up for the next t tosses is exactly the same as the probability for no "Head" to turn up for the first t tosses of the game.

This characteristic property illustrates the close relationship between the Geometric Distribution and the Exponential Distribution. This relationship is made explicit in the interactive animation.

In fact, the Geometric Distribution may be perceived as the discrete version of the Exponential Distribution. This link can be formalized as follows. Suppose that the delay between two consecutive tosses is made to tend toward 0. Then it can be shown that the distribution function of the Geometric Distribution converges towards that of an exponential distribution.

Note that we calculate :
       * The mean (here),
       * The moment generating function (here),
of the geometric distribution by methods that involve only the memoryless property and conditional expectations. 

Alternative definition

The Geometric Distribution is sometimes presented as that of the number of tosses before the first "Head" turns up (instead of the number of tosses needed to obtain the first "Head"). Let L' denote this new r.v.. We obviously have :

L' = L - 1

As an exercise, you may derive the basic properties of this r.v. :

    * Either by direct calculation,

    * Or by using the results relative to the functions of r.v., as L'  is just a translation of L (see here).

Geometric distribution and negative binomial distribution

The geometric distribution is a special case of a more general distribution called the "negative binomial distribution". Because it is quite a bit simpler than the negative binomial distribution, we give the geometric distribution a separate treatment.

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These results are demonstrated in this Tutorial.

We also clarify the relationship between the geometric and the exponential distributions by showing that a geometric r.v. may be considered as resulting from the discretization of an exponential r.v. (The reverse relation is addressed here).

THE GEOMETRIC DISTRIBUTION

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