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Geometric DistributionEdit

Probability mass function
Cumulative distribution function
Parameters   success probability (real)
Median   (not unique if   is an integer)
Ex. kurtosis  
MGF  ,

There are two similar distributions with the name "Geometric Distribution".

  • The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}
  • The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... }

These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one. We will use X and Y to refer to distinguish the two.


The shifted Geometric Distribution refers to the probability of the number of times needed to do something until getting a desired result. For example:

  • How many times will I throw a coin until it lands on heads?
  • How many children will I have until I get a girl?
  • How many cards will I draw from a pack until I get a Joker?

Just like the Bernoulli Distribution, the Geometric distribution has one controlling parameter: The probability of success in any independent test.

If a random variable X is distributed with a Geometric Distribution with a parameter p we write its probability mass function as:


With a Geometric Distribution it is also pretty easy to calculate the probability of a "more than n times" case. The probability of failing to achieve the wanted result is  .

Example: a student comes home from a party in the forest, in which interesting substances were consumed. The student is trying to find the key to his front door, out of a keychain with 10 different keys. What is the probability of the student succeeding in finding the right key in the 4th attempt?



The probability mass function is defined as:




Let q=1-p


We can now interchange the derivative and the sum.



We derive the variance using the following formula:


We have already calculated E[X] above, so now we will calculate E[X2] and then return to this variance formula:


Let q=1-p


We now manipulate x2 so that we get forms that are easy to handle by the technique used when deriving the mean.


We then return to the variance formula


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