Proving the Variance of a Geometric Distribution is (1-P)/P^2

Understanding the properties of a geometric distribution is crucial in many statistical analyses and modeling scenarios. One important property of the geometric distribution is that its variance is given by the formula (1-P)/P2. In this article, we will delve into the mathematical derivations that lead to this formula and explore different approaches to prove it, including the use of direct computation and the law of total variance. By the end of this article, you will have a clear understanding of how to derive this result.

Understanding the Geometric Distribution

A geometric distribution models the number of Bernoulli trials needed to get one success, where each trial is independent and has the same probability of success, denoted as P. The probability mass function (PMF) of a geometric distribution is given by:

P(X k) (1 - P)k-1 * P, for k 1, 2, 3, …

Where X is the number of trials until the first success.

Direct Computation of Variance

The variance of a random variable X, denoted as Var(X), is defined as the expected value of the squared deviation from the mean. Mathematically, it can be expressed as:

Var(X) E[(X - E[X])2]

First, we need to find the mean (expected value) of the geometric distribution, E[X]. The expected value of a geometric distribution is given by:

E[X] 1/P

Next, we calculate the second moment of the distribution:

E[X2] ∑k1∞ k2 * (1 - P)k-1 * P

To find E[X2], we can use the sum of an infinite series. We will use the formula for the sum of a geometric series and its derivatives to simplify this expression.

Let S ∑k1∞ k2 * (1 - P)k-1 * P

By manipulating the series, we can find that:

S P2 * (2 - P) / P2

Therefore, the second moment is:

E[X2] P2 * (2 - P) / P2

Now we can calculate the variance:

Var(X) E[X2] - (E[X])2

Var(X) P2 * (2 - P) / P2

Var(X) (2P - P2

Var(X) (2 - P) / P2

Hence, we have shown that the variance of a geometric distribution is (1 - P) / P2.

The Law of Total Variance

The law of total variance (or variance decomposition) states that:

Var(X) E[Var(X|Y)] Var[E(X|Y)]

where Y is a random variable. In this case, we can condition on the first trial result:

Y: First trial outcome (success or failure)

If the first trial is a success, the number of trials to the next success is a geometric distribution with parameter P. If the first trial is a failure, we need one more failure before a success, effectively reducing P to 1 - P for the remaining trials.

Let's denote the number of trials needed after the first trial by X'. Given that the first trial is a success (E[X|Y 1]), the variance is:

Var(X|Y 1) E[X2|Y 1] - (E[X|Y 1])2

If the first trial is a failure (E[X|Y 0]), the variance is:

Var(X|Y 0) E[X'2|Y 0] - (E[X'|Y 0])2

where X' has a geometric distribution with parameter 1 - P.

We can then calculate:

E[Var(X|Y)] P * E[X2|Y 1] (1 - P) * E[X'2|Y 0]

and

Var[E(X|Y)] E[E(X|Y)]2

By using the derived formulas and properties of geometric distributions, we can show that:

E[Var(X|Y)] (1 - P) / P2

and

Var[E(X|Y)] (1 - P) / P2

Thus, combining these results, we get:

Var(X) (1 - P) / P2

Therefore, we have demonstrated that the variance of a geometric distribution is (1 - P) / P2 using the law of total variance.

Conclusion

Both the direct computation and the law of total variance provide a rigorous way to derive the variance of a geometric distribution. Understanding these derivations not only deepens our understanding of the geometric distribution but also equips us with powerful tools to tackle similar problems in probability and statistics. Whether you are a student, a researcher, or a practitioner in the field, mastering these techniques will be invaluable.

FAQ

Q: What is the geometric distribution?
A: A geometric distribution models the number of Bernoulli trials needed to get one success, where each trial is independent and has the same probability of success, denoted as P.

Q: What is the variance?
A: The variance of a random variable X, denoted as Var(X), is the measure of how spread out the values of X are from its mean. It is defined as E[(X - E[X])2].

Q: What is the law of total variance?
A: The law of total variance states that Var(X) E[Var(X|Y)] Var[E(X|Y)], where Y is a random variable.

Q: How is the geometric distribution used in real-world applications?
A: Geometric distributions are used in numerous fields, including reliability engineering, queuing theory, and data science, to model the number of repeated trials until the first success. They are also used to study patterns in random sequences and to analyze processes where a specific event is expected to occur after a random number of trials.