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: \[X \sim Geo(p)\]
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}#: \[X\sim Geo\bigg(\cfrac{1}{6}\bigg)\]
#\phantom{0}#
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.
Compute #\mathbb{P}(X = 3)#. Round your answer to #3# decimal places.
There are a number of different ways we can calculate #\mathbb{P}(X = 3)#. Click on one of the panels to toggle a specific solution.
To calculate #\mathbb{P}(X = 3)# in Excel, make use of the following function:
If we observe the first success on the third trial #(X = 3)#, then we will have observed #2# failures and #1# success.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)
Thus, to calculate #\mathbb{P}(X = 3)#, we run the following command:
\[= \text{NEGBINOM.DIST}(2, 1, 0.1, \text{FALSE})\]
This gives:
\[\mathbb{P}(X = 3) = 0.081\]
To calculate #\mathbb{P}(X = 3)# in R, make use of the following function:
If we observe the first success on the third trial #(X = 3)#, then we will have observed #2# failures.dgeom(x, prob)
- x: The number of failures before a success occurs.
- prob: The probability of success for each trial.
Thus, to calculate #\mathbb{P}(X = 3)#, we run the following command:
\[\text{dgeom}(x = 2, prob = 0.1)\]
This gives:
\[\mathbb{P}(X = 3) = 0.081\]
#\phantom{0}#
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.
There are a number of different ways we can calculate #\mathbb{P}(X \leq 7)#. Click on one of the panels to toggle a specific solution.
To calculate #\mathbb{P}(X \leq 7)# in Excel, make use of the following function:
If we observe the first success on the seventh trial or ealier #(X \leq 7)#, then we will have observed at most #6# failures and #1# success.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)
Thus, to calculate #\mathbb{P}(X \leq 7)#, we run the following command:
\[= \text{NEGBINOM.DIST}(6, 1, 0.2, \text{TRUE})\]
This gives:
\[\mathbb{P}(X \leq 7) = 0.790\]
To calculate #\mathbb{P}(X \leq 7)# in R, make use of the following function:
If we observe the first success on the third trial or earlier #(X \leq 7)#, then we will have observed at most #6# failures.pgeom(q, prob)
- q: The number of failures before a success occurs.
- prob: The probability of success for each trial.
Thus, to calculate #\mathbb{P}(X \leq 7)#, we run the following command:
\[\text{pgeom}(q = 6, prob = 0.2)\]
This gives:
\[\mathbb{P}(X \leq 7) = 0.790\]
#\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:\[\sigma = \sqrt{\cfrac{1-p}{p^2}}\]
The expected value of a geometric random variable #X\sim Geo(p)# is calculated as follows:
\[\begin{array}{rcl}
\mu&=& \cfrac{1}{p} \\\\
&=& \cfrac{1}{0.30} \\\\
&=& 3.33
\end{array}\]
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