The exact formula for the standard deviation of a geometric distribution is sqrt((1-p)/p).
Understanding the Geometric Distribution
A geometric distribution models the number of independent trials needed to achieve the first success in a sequence of Bernoulli trials. Each trial has only two possible outcomes: success or failure. The probability of success, denoted by p, remains constant for every trial. The random variable in a geometric distribution is the number of trials Y required to get the first success.
The Formula for Standard Deviation
The standard deviation measures the spread or dispersion of the distribution. For a geometric random variable Y representing the number of trials to the first success with a probability of success p on each trial, the standard deviation is given by the formula:
Standard Deviation (σ) = sqrt((1-p)/p)
As stated in the reference: "The standard deviation of a geometric random variable Y, which represents the number of trials needed to achieve the first success with probability p of success on each trial, is given by: standard deviation = σy = sqrt((1-p)/p)".
Here's a breakdown of the components:
- σ (sigma): Represents the standard deviation.
- p: The probability of success on a single trial. This value must be between 0 and 1 (0 < p ≤ 1).
- 1-p: The probability of failure on a single trial, sometimes denoted as q.
- sqrt(): The square root function.
Key Characteristics
Here are some key characteristics related to the standard deviation and the geometric distribution:
- The standard deviation increases as the probability of success (p) decreases. This means if success is rare, the number of trials needed to get the first success can vary widely.
- The standard deviation is smaller when the probability of success (p) is higher, indicating less variability in the number of trials required.
Example
Suppose you are flipping a biased coin where the probability of getting heads (success) is p = 0.25. What is the standard deviation of the number of flips needed to get the first head?
Using the formula:
σ = sqrt((1-p)/p)
σ = sqrt((1-0.25)/0.25)
σ = sqrt(0.75/0.25)
σ = sqrt(3)
σ ≈ 1.732
So, the standard deviation is approximately 1.732 trials.
