Geometric Means, Variances, and Tail Probabilities

Two Ways to Count the Same Wait

A geometric random variable describes how long a sequence of independent trials continues until the first success. The numerical answer depends on what is counted: the failures before that success, or all the trials up to and including it. These are two measurements of the same wait, but their means differ by one. Their variances and appropriately translated tail probabilities agree.

Suppose each trial succeeds with probability \(p\), where \(0<p<1\), and write \(q=1-p\) for the failure probability. The trials are independent and have the same success probability. The probability mass functions below follow by requiring an initial sequence of failures and then a success.

The shift \(Y=X+1\) is the key to comparing the two conventions. We will also derive the moments directly, because the calculations make clear where the variance comes from.

Means and Variances

The geometric series and its derivatives give the sums needed for the moments. For \(0<q<1\), the power series \(\sum_{k=0}^{\infty}q^k=1/(1-q)\) can be differentiated term by term inside its radius of convergence. Differentiating once and then differentiating again yields

For the second identity, the first differentiation gives \(\sum_{k=1}^{\infty}kq^{k-1}=(1-q)^{-2}\). Differentiating that identity gives \(\sum_{k=2}^{\infty}k(k-1)q^{k-2}=2(1-q)^{-3}\). Multiplying by \(q\) and adding the first identity gives the displayed sum of \(k^2q^{k-1}\).

Proof. Since \(1-q=p\), the first series identity gives

Similarly, the second identity gives the second raw moment:

Therefore,

Because \(Y=X+1\), linearity of expectation gives \(E[Y]=E[X]+1=q/p+1=1/p\), using \(p+q=1\). Adding a constant does not change variance, so \(\operatorname{Var}(Y)=\operatorname{Var}(X)=q/p^2\). This also proves the stated formulas for \(Y\). \(\square\)

The mean counts include the difference in convention: the trials-through-success mean is exactly one larger. The variance describes spread around the mean, and shifting every possible value by one leaves that spread unchanged.

Survival Probabilities and Integer Thresholds

A survival probability is the probability that a waiting time reaches or exceeds a specified threshold. For a discrete variable, it is important to distinguish “at least \(k\)” from “more than \(k\).” Summing the mass function gives concise formulas for both conventions.

Proof. For \(k\geq0\), the event \(X\geq k\) means that the first \(k\) trials all fail. Summing the mass function confirms the same result:

For \(m\geq0\), \(Y>m\) means the first \(m\) trials all fail. Equivalently, since \(Y=X+1\), it is the event \(X\geq m\). Thus \(P(Y>m)=q^m\), and taking the complement gives \(P(Y\leq m)=1-q^m\). Finally, \(Y\geq r\) is the same event as \(Y>r-1\), so \(P(Y\geq r)=q^{r-1}\). \(\square\)

These formulas pair naturally: \(P(X\geq k)=P(Y>k)=q^k\). The events match because \(Y=X+1\), not because the two variables use the same threshold and inequality.

Common Pitfall and Interpretation

A frequent error is to use \(q^k\) for \(P(Y\geq k)\). That exponent is one too large for this event: \(Y\geq k\) allows success on trial \(k\), so only the first \(k-1\) trials are required to be failures. The correct expression is \(P(Y\geq k)=q^{k-1}\). In contrast, \(P(Y>k)=q^k\), because success must not occur on any of the first \(k\) trials.

A related mistake is to add one to the variance when changing conventions. The identity \(Y=X+1\) increases the mean by one but does not increase the variance. The same shift also helps translate events: \(X\geq k\) and \(Y>k\) describe the same waiting outcomes, while \(Y\geq k\) corresponds to \(X\geq k-1\).

Means summarize the typical wait, and survival probabilities answer questions about unusually long waits or required trial limits. Both are useful in planning repeated procedures: an expected wait alone does not tell you the probability of exceeding a particular deadline. The geometric formulas provide that probability directly once the success chance and counting convention are clear.