The Geometric Probability Distribution

Chapter 4. Probability Distributions: Discrete Probability Distributions

The Geometric Probability Distribution

Geometric Experiment
In a geometric experiment:

We conduct an unknown number of independent Bernoulli trials.
The probability of success #p# is the same for each trial.
The variable of interest #X# is the total number of trials needed until we observe a success for the first time.

Geometric Distribution
Let #X# be the number of trials needed to observe the first success in a geometric experiment, then #X# is a geometric random variable with range: \[R(X)=\{1,2,\ldots\}\]

We say that #X# is geometrically distributed with parameter #p# and write this as: \[X \sim Geo(p)\]

GeoGebra

Suppose we roll a regular six-sided die and define a success as rolling a #6#.

Let #X# be the number of attempts needed to roll a #6#.

Then #X# is geometrically distributed with #p=\mathbb{P}(\text{Roll a }6) = \cfrac{1}{6}#: \[X\sim Geo\bigg(\cfrac{1}{6}\bigg)\]

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Computation of Geometric Probabilities with Statistical Software
Let #X# be a geometric random variable with parameter #p#.

To compute #\mathbb{P}(X=x)# in Excel, make use of the following function:

NEGBINOM.DIST(number_f, number_s, probability_s, cumulative)

number_f: The number of failures.

number_s: The number of successes.

NOTE: When using this function to perform geometric probability calculations, this argument will always be set to 1.

probability_s: The probability of success for each trial.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution (at most number_f failures)

FALSE - uses the probability mass function (exactly number_f failures)

To compute #\mathbb{P}(X=x)# in R, make use of the following function:

dgeom(x, prob)

x: The number of failures before a success occurs.

prob: The probability of success for each trial.

Suppose #X# is a geometrically distributed random variable with #p=0.4#.

Compute #\mathbb{P}(X = 7)#. Round your answer to #3# decimal places.

#\mathbb{P}(X = 7)=0.019#

There are a number of different ways we can calculate #\mathbb{P}(X = 7)#. Click on one of the panels to toggle a specific solution.

Excel Calculation

To calculate #\mathbb{P}(X = 7)# in Excel, make use of the following function:

NEGBINOM.DIST(number_f, number_s, probability_s, cumulative)

number_f: The number of failures.

number_s: The number of successes.

probability_s: The probability of success for each trial.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution (at most number_f failures)

FALSE - uses the probability mass function (exactly number_f failures)

If we observe the first success on the seventh trial #(X = 7)#, then we will have observed #6# failures and #1# success.

Thus, to calculate #\mathbb{P}(X = 7)#, we run the following command:
\[= \text{NEGBINOM.DIST}(6, 1, 0.4, \text{FALSE})\]
This gives:
\[\mathbb{P}(X = 7) = 0.019\]

R Calculation

To calculate #\mathbb{P}(X = 7)# in R, make use of the following function:

dgeom(x, prob)

x: The number of failures before a success occurs.

prob: The probability of success for each trial.

If we observe the first success on the seventh trial #(X = 7)#, then we will have observed #6# failures.

Thus, to calculate #\mathbb{P}(X = 7)#, we run the following command:
\[\text{dgeom}(x = 6, prob = 0.4)\]
This gives:
\[\mathbb{P}(X = 7) = 0.019\]

New example

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Computation of Cumulative Geometric Probabilities with Statistical Software
Let #X# be a geometric random variable with parameter #p#.

To calculate cumulative probabilities for a geometric distribution in Excel, make use of the following function:

NEGBINOM.DIST(number_f, number_s, probability_s, cumulative)

number_f: The number of failures.

number_s: The number of successes.

NOTE: When using this function to perform geometric probability calculations, this argument will always be set to 1.

probability_s: The probability of success for each trial.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution (at most number_f failures)

FALSE - uses the probability mass function (exactly number_f failures)

To calculate cumulative probabilities for a geometric distribution in R, make use of the following function:

pgeom(q, prob)

q: The number of failures before a success occurs.

prob: The probability of success for each trial.

Cumulative Properties of Discrete Distributions
Since the geometric distribution is discrete, the following properties regarding its cumulative probabilities hold:

#\mathbb{P}(X \leq x) = \mathbb{P}(X=0) +\mathbb{P}(X=1) + \ldots + \mathbb{P}(X=x)#
#\mathbb{P}(X \lt x) = \mathbb{P}(X \leq x-1)#
#\mathbb{P}(X \gt x) = 1 - \mathbb{P}(X \leq x)#
#\mathbb{P}(X \geq x) = 1 - \mathbb{P}(X \leq x-1)#

Additionally, for any #2# values #a# and #b# in the range of #X# (with #a<b#):

#\mathbb{P}(a \leq X \leq b) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a - 1)#
#\mathbb{P}(a \leq X \lt b) = \mathbb{P}(X \leq b - 1) - \mathbb{P}(X \leq a - 1)#
#\mathbb{P}(a \lt X \leq b) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a)#
#\mathbb{P}(a \lt X \lt b) = \mathbb{P}(X \leq b - 1) - \mathbb{P}(X \leq a)#

Suppose #X# is a geometrically distributed random variable with #p=0.2#.

Compute #\mathbb{P}(X \leq 6)#. Round your answer to #3# decimal places.

#\mathbb{P}(X \leq 6)=0.738#

There are a number of different ways we can calculate #\mathbb{P}(X \leq 6)#. Click on one of the panels to toggle a specific solution.

Excel Calculation

To calculate #\mathbb{P}(X \leq 6)# in Excel, make use of the following function:

NEGBINOM.DIST(number_f, number_s, probability_s, cumulative)

number_f: The number of failures.

number_s: The number of successes.

probability_s: The probability of success for each trial.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution (at most number_f failures)

FALSE - uses the probability mass function (exactly number_f failures)

If we observe the first success on the sixth trial or ealier #(X \leq 6)#, then we will have observed at most #5# failures and #1# success.

Thus, to calculate #\mathbb{P}(X \leq 6)#, we run the following command:
\[= \text{NEGBINOM.DIST}(5, 1, 0.2, \text{TRUE})\]
This gives:
\[\mathbb{P}(X \leq 6) = 0.738\]

R Calculation

To calculate #\mathbb{P}(X \leq 6)# in R, make use of the following function:

pgeom(q, prob)

q: The number of failures before a success occurs.

prob: The probability of success for each trial.

If we observe the first success on the eighth trial or earlier #(X \leq 6)#, then we will have observed at most #5# failures.

Thus, to calculate #\mathbb{P}(X \leq 6)#, we run the following command:
\[\text{pgeom}(q = 5, prob = 0.2)\]
This gives:
\[\mathbb{P}(X \leq 6) = 0.738\]

New example

#\text{}#

Mean, Variance, and Standard Deviation of a Geometric Random Variable
The expected value of a geometric random variable is calculated with the following formula: \[\mu = \cfrac{1}{p}\]

The variance of a geometric random variable is calculated with the following formula:\[\sigma^2 = \cfrac{1-p}{p^2}\]

The standard deviation of a geometric random variable is calculated with the following formula:\[\sigma = \sqrt{\cfrac{1-p}{p^2}}\]

Let #X# be a geometrically distributed random variable with parameter #p=0.25#.

Calculate the expected value of #X#. Round your answer to #2# decimal places.

#\mu=4#

The expected value of a geometric random variable #X\sim Geo(p)# is calculated as follows:
\[\begin{array}{rcl}
\mu&=& \cfrac{1}{p} \\\\
&=& \cfrac{1}{0.25} \\\\
&=& 4
\end{array}\]

New example