How to Find the Moments of the Geometric Distribution - dummies

How to Find the Moments of the Geometric Distribution

By Alan Anderson

Moments are summary measures of a probability distribution, and include the expected value, variance, and standard deviation. The moments of the geometric distribution depend on which of the following situations is being modeled:

  • The number of trials required before the first success takes place

  • The number of failures that occur before the first success

Just as with the binomial distribution, the geometric distribution has a series of simplified formulas for computing these moments.

How to calculate the expected value of the geometric distribution

The expected value of the geometric distribution when determining the number of trials required until the first success is

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The expected value of the geometric distribution when determining the number of failures that occur before the first success is

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For example, when flipping coins, if success is defined as “a heads turns up,” the probability of a success equals p = 0.5; therefore, failure is defined as “a tails turns up” and 1 – p = 1 – 0.5 = 0.5. On average, there’ll be (1 – p)/p = (1 – 0.5)/0.5 = 0.5/0.5 = 1 tails before the first heads turns up.

Notice how the two results provide the same information; it takes an average of two flips to get the first heads, or on average there should be one tails before the first heads turns up.

How to compute the variance and standard deviation of the geometric distribution

The variance and standard deviation of the geometric distribution when determining the number of trials required until the first success or when determining the number of failures that occur before the first success are

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For example, suppose you flip a coin until the first heads turns up. The expected number of trials required until the first heads turns up is

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The variance is

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