Geometric Distribution

ABSTRACT

The Geometric Distribution models the number of independent Bernoulli Trials required to achieve the first success. It is defined over an infinite discrete sample space.

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Definition

Suppose we conduct a sequence of independent trials, each with a probability of success $p$ and probability of failure $q=(1-p)$. The random variable $X$ represents the number of the trial on which the first success occurs.

Probability Mass Function (PMF)

For $k\in \{1,2,3,\dots \}$:

$$\boxed{f(k)=P(X=k)=(1-p{)}^{k-1}p}$$

• $(1-p{)}^{k-1}$: The probability of having $k-1$ consecutive failures.
• $p$: the probability that the $k^{th}$ trial is a success.

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Proof: Sum of Probabilities

By the Law of Total Probability, the sum of all probabilities in the sample space $S={\mathbb{Z}}^{+}$ must equal $1$.

Using the formula for an infinite Geometric Series where the first term $a_1=1$ and the common ratio $r=(1-p)$:

$$\begin{array}{rl}\sum _{k=1}^{\infty }(1-p{)}^{k-1}\cdot p& =p\cdot \sum _{k=1}^{\infty }(1-p{)}^{k-1}\\ & =p\cdot \left(\frac{1}{1-(1-p)}\right)\\ & =p\cdot \left(\frac{1}{p}\right)\\ & =\boxed{1}\end{array}$$

NOTE

The series converges because $0<p<1$, which implies $|1-p|<1$.

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Properties

Property	Value
Sample Space	$S=\{1,2,3,\dots \}$
Expected Value	$E[X]=\frac{1}{p}$
Memoryless Property	$P(X>n+k\mid X>n)=P(X>k)$

The Memoryless Property

The Geometric distribution is unique among discrete distributions because it is memoryless. This means that if you have already failed $n$ times, the probability of needing $k$ more trials is the same as the probability of needing $k$ trials at the very start. The “coin has no memory” of previous failures.

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Example: Rolling a Die

What is the expected number of rolls to get a 6 on a fair die?

• Here, $p=1/6$.
• Using the expectation formula: $E[X]=\frac{1}{1/6}=\boxed{6}$ rolls.

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Related Notes

• Binomial Distribution: Contrast this with the Binomial, where the number of trials $n$ is fixed and we count successes.
• Case Analysis: Often used to derive the Expected Value of a geometric distribution via “First-Step Analysis.”
• Expected Value: The long-run average of trials needed for success.