The variance of a geometric random variable is given by the formula (1 - p) / p².
Understanding the geometric distribution is key to grasping its variance. A geometric random variable represents the number of Bernoulli trials needed to get the first success, where each trial is independent and has a constant probability of success, denoted by p. The variance measures the spread or dispersion of this random variable around its mean.
The Formula Explained
Based on the provided reference, the variance (v) of a geometric random variable is explicitly stated as:
$$v = \frac{1 - p}{p^2}$$
Here's a breakdown:
- v: Represents the variance of the geometric distribution.
- p: Represents the probability of success on any single trial (0 < p ≤ 1).
- 1 - p: Represents the probability of failure on any single trial.
- p²: Represents the square of the probability of success.
The reference notes: "Notice that the mean m is ( 1 - p ) / p and the variance v is ( 1 - p ) / p 2." This confirms the formula provided.
Why is the Variance Important?
The variance tells us how much the number of trials until the first success can vary from the expected number of trials (the mean).
- Higher Variance: Indicates that the results are more spread out, meaning you might need significantly more or fewer trials than the mean to achieve the first success. This typically occurs when p is small.
- Lower Variance: Indicates that the results are clustered more closely around the mean, suggesting the number of trials needed is more predictable. This happens when p is large.
Example
Let's say you are flipping a biased coin where the probability of getting heads (p) is 0.25. The number of flips needed to get the first head follows a geometric distribution.
- Probability of success (p) = 0.25
- Probability of failure (1 - p) = 1 - 0.25 = 0.75
Using the variance formula:
Variance = (1 - p) / p² = 0.75 / (0.25)² = 0.75 / 0.0625 = 12
So, for this biased coin, the variance in the number of flips needed to get the first head is 12. This suggests a relatively large spread in the possible outcomes.
Summary Table
| Measure | Formula | Description |
|---|---|---|
| Mean (m) | (1 - p) / p | Expected number of trials until first success |
| Variance (v) | (1 - p) / p² | Spread of the distribution |
Understanding the variance helps in quantifying the uncertainty associated with waiting for the first success in a sequence of independent Bernoulli trials.
