The expected value of a geometric distribution depends on which geometric random variable definition is used. Both versions model repeated independent Bernoulli trials with constant success probability \(p\), where \(0 \lt p \lt 1\).
- Trials-until-success version: \(X \in \{1,2,3,\dots\}\) is the number of trials needed to get the first success. \[ P(X=k)=p \cdot (1-p)^{k-1}. \]
- Failures-before-success version: \(W \in \{0,1,2,\dots\}\) is the number of failures before the first success. \[ P(W=w)=p \cdot (1-p)^w. \]
Result: expected value of a geometric distribution
The two means differ by exactly 1 because \(X=W+1\). Therefore, \[ \mathbb{E}[X]=\mathbb{E}[W]+1. \]
Derivation for \(X\): number of trials until first success
By definition, \[ \mathbb{E}[X]=\sum_{k=1}^{\infty} k \cdot P(X=k)=\sum_{k=1}^{\infty} k \cdot p \cdot (1-p)^{k-1}. \] Let \(q=1-p\). Then \[ \mathbb{E}[X]=p \cdot \sum_{k=1}^{\infty} k \cdot q^{k-1}. \]
Use the geometric series identity (valid for \(|q| \lt 1\)): \[ \sum_{k=0}^{\infty} q^k=\frac{1}{1-q}. \] Differentiate both sides with respect to \(q\): \[ \sum_{k=1}^{\infty} k \cdot q^{k-1}=\frac{1}{(1-q)^2}. \]
Substitute into the expectation: \[ \mathbb{E}[X]=p \cdot \frac{1}{(1-q)^2}. \] Since \(1-q=1-(1-p)=p\), it follows that \[ \mathbb{E}[X]=p \cdot \frac{1}{p^2}=\frac{1}{p}. \]
Derivation for \(W\): number of failures before first success
Since \(W=X-1\), \[ \mathbb{E}[W]=\mathbb{E}[X]-1=\frac{1}{p}-1=\frac{1-p}{p}. \] This agrees with the direct summation from \(P(W=w)=p \cdot (1-p)^w\) as well.
Worked example (numerical expected value)
If \(p=0.20\), then the expected value of a geometric distribution (trials-until-success) is \[ \mathbb{E}[X]=\frac{1}{0.20}=5. \] Interpreted in context: on average, 5 trials are needed to obtain the first success when each trial succeeds with probability 0.20.
| Definition | Support | PMF | Expected value |
|---|---|---|---|
| \(X\): trials until first success | \(\{1,2,3,\dots\}\) | \(P(X=k)=p \cdot (1-p)^{k-1}\) | \(\mathbb{E}[X]=1/p\) |
| \(W\): failures before first success | \(\{0,1,2,\dots\}\) | \(P(W=w)=p \cdot (1-p)^w\) | \(\mathbb{E}[W]=(1-p)/p\) |
Visualization: PMF bars and the mean \(1/p\)
Common related quantity (often reported)
The variance is frequently paired with the expected value: \[ \mathrm{Var}(X)=\frac{1-p}{p^2} \quad \text{and} \quad \mathrm{Var}(W)=\frac{1-p}{p^2}. \] The standard deviation is the square root of variance.
Assumptions behind the geometric model
- Trials are independent.
- Each trial has the same success probability \(p\).
- The first success ends the count, creating a geometric waiting-time distribution.