What is the mean of the negative binomial distribution under the total-trials convention?

Conditions

• Using the total-trials convention (counting all trials).
• Success probability $p > 0$.

Reasoning, step by step

1. Identify the convention: counting total trials until r successes.
2. Apply the formula for the mean: r divided by p.

Example

With $r=3$ and $p=0.09$, the mean is $3/0.09 = 33$⅓ trials.

Common misconceptions

• Confusing the total-trials mean with the failures-only mean.
• Thinking the mean is $r*p$.

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Explore next

• How does the failures-only convention affect the mean?
• What is the variance of this distribution?