How does the failures-only convention differ from the total-trials convention in the negative binomial model?

Conditions

• Comparing two different definitions of the negative binomial random variable.

Reasoning, step by step

1. Define the total-trials variable: counts all trials until r successes.
2. Define the failures-only variable: counts only failures until r successes.
3. Note that Total Trials = Failures + Successes.
4. Conclude that the failures-only mean is ($r/p$) - r, or equivalently, the variable is shifted by r.

Example

The script states: 'A failures-only convention shifts the variable and its mean by r.'

Common misconceptions

• Thinking the two conventions yield the same numerical value for the random variable.
• Forgetting that the shift amount is exactly r.

Watch the explanation

Explore next

• What is the mean under the failures-only convention?
• Why are there different conventions?