Geometric distribution - Revision history

DavidB4-bot: /* top */Spelling/Grammar Check, typos fixed: Therefore → Therefore,

2016-07-18T17:59:00Z

top: Spelling/Grammar Check, typos fixed: Therefore → Therefore,

DavidB4-bot: /* Probability-generating function */clean up & uniformity

2016-07-13T13:23:15Z

Probability-generating function: clean up & uniformity

RonLar at 13:43, August 14, 2010

2010-08-14T13:43:37Z

RonLar: /* Probability-generating function */ def. of prob.gen.fct.

2010-08-09T19:11:36Z

Probability-generating function: def. of prob.gen.fct.

RonLar: probability generating function

2010-08-07T11:46:49Z

probability generating function

RonLar: /* Mean and Variance */

2010-08-03T17:35:11Z

Mean and Variance

RonLar: /* Mean */

2010-08-03T16:46:44Z

Mean

RonLar: mean

2010-07-28T15:08:42Z

mean

RonLar: a start...

2010-07-27T13:41:17Z

a start...

New page

The '''geometric distribution''' is a discrete distribution which describes the
number of repetitions of a [[Bernoulli experiment]] to get a success. Therefore
its support are the positive integers, {1,2,3,…}

== Alternative definition ==
Some authors describe the geometric distribution as the number of repetitions
until the first success, its support being the non-negative integers
{0,1,2,…}[[category:Probability and Statistics]]

← Older revision		Revision as of 13:23, July 13, 2016
Line 29:		Line 29:


−	[[~~category~~:Probability and Statistics]]	+	[[Category:Probability and Statistics]]

@@ Line 20: / Line 20: @@
 == Probability-generating function ==
-The [[probability-generating function]] <math>G_X(z)</math> is:
+The [[probability-generating function]] <math>G_X(z) = \sum_{n=0}^\infty z^n \cdot \mathbf{P}[X=n]</math> is:
 <math>G_X(z) = \frac{z\,p}{1-z(1-p)}</math>.

← Older revision		Revision as of 11:46, August 7, 2010
Line 18:		Line 18:

	<math>Var(X) = \frac{1-p}{p^2}</math>.		<math>Var(X) = \frac{1-p}{p^2}</math>.
		+
		+	== Probability-generating function ==
		+	The [[probability-generating function]] <math>G_X(z)</math> is:
		+
		+	<math>G_X(z) = \frac{z\,p}{1-z(1-p)}</math>.

@@ Line 15: / Line 15: @@
 p</math><math>=\frac{p}{(1-q)^2}= \frac{1}{p}</math>
-Similarly, the variance can be calculated:
+The variance can be calculated similarly:
-<math>Var(X) = \frac{1-p}{p^2}</math>
+<math>Var(X) = \frac{1-p}{p^2}</math>.

@@ Line 8: / Line 8: @@
 {0,1,2,…}[[category:Probability and Statistics]]
-== Mean  ==
+== Mean and Variance ==
 The [[mean]] for a [[random variable]] ''X'' following a geometric distribution
 with a probability of success ''p'' (and ''q'' = 1 - ''p'') is
@@ Line 14: / Line 14: @@
 <math>\mathbf{E}[X] = \sum_{n=0}^{\infty} (n+1) q^n
 p</math><math>=\frac{p}{(1-q)^2}= \frac{1}{p}</math>

@@ Line 1: / Line 1: @@
 The '''geometric distribution''' is a discrete distribution which describes the
-number of repetitions of a [[Bernoulli experiment]] to get a success. Therefore
+number of repetitions of a [[Bernoulli experiment]] to get a success. Therefore,
 its support are the positive integers, {1,2,3,…}

← Older revision		Revision as of 15:08, July 28, 2010
Line 7:		Line 7:
	until the first success, its support being the non-negative integers		until the first success, its support being the non-negative integers
	{0,1,2,…}[[category:Probability and Statistics]]		{0,1,2,…}[[category:Probability and Statistics]]
		+
		+	== Mean ==
		+	The [[mean]] for a [[random variable]] ''X'' following a geometric distribution
		+	with a probability of success ''p'' (and ''q'' = 1 - ''p'') is
		+
		+	<math>\mathbf{E}[X] = \sum_{n=0}^{\infty} (n+1) q^n
		+	p</math><math>=\frac{p}{(1-q)^2}= \frac{1}{p}</math>