VISUAL LESSON · 4 PICTURES

Negative Binomial: Wait for the r-th Success

How many independent Bernoulli trials are needed to reach the r-th success?

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BEFORE THE FIRST PICTURE

How this lesson unfolds.

Begin with Fixed trials or wait for a success? The sequence then connects The last trial must be the r-th success and builds toward Independent waiting-time variances add.

Choose the model by what stops the experiment. Binomial fixes n trials; geometric waits for the first success; negative binomial waits for the r-th success. — Picture 1 of 4 · tap to inspect in full resolution

Fixed trials or wait for a success?

INTUITION

Binomial fixes n trials; geometric waits for the first success; negative binomial waits for the r-th success.

Learning goalExplain model choice for the negative binomial distribution.
Helpful foundationsBernoulli trials · binomial distribution · geometric distribution
Common trapNegative binomial fixes the required success count, not the number of trials.
Next connectionThe last trial must be the r-th success

The last trial must be the r-th success. To finish on trial x, place r-1 successes among the first x-1 trials and require the final trial to be a success. — Picture 2 of 4 · tap to inspect in full resolution

Why does the PMF use x-1 choose r-1?

INTUITION

To finish on trial x, place r-1 successes among the first x-1 trials and require the final trial to be a success.

Learning goalExplain pmf example for the negative binomial distribution.
Helpful foundationsBernoulli trials · binomial distribution · geometric distribution
Common trapNegative binomial fixes the required success count, not the number of trials.
Next connectionThe total wait is a sum of geometric waits

The total wait is a sum of geometric waits. Write X=G1+...+Gr, where each Gi is the independent waiting time to the next success; linearity gives E[X]=r/p. — Picture 3 of 4 · tap to inspect in full resolution

Why is the mean r divided by p?

INTUITION

Write X=G1+...+Gr, where each Gi is the independent waiting time to the next success; linearity gives E[X]=r/p.

MATHEMATICAL STATEMENT

Write X=G1+...+Gr, where each Gi is the independent waiting time to the next success; linearity gives E[X]=r/p.

Learning goalExplain mean derivation for the negative binomial distribution.
Helpful foundationsBernoulli trials · binomial distribution · geometric distribution
Common trapNegative binomial fixes the required success count, not the number of trials.
Next connectionIndependent waiting-time variances add

Independent waiting-time variances add. Each geometric wait has variance (1-p)/p squared; independence removes covariance terms and r equal variances add. — Picture 4 of 4 · tap to inspect in full resolution

Why is the variance r(1-p)/p squared?

INTUITION

Each geometric wait has variance (1-p)/p squared; independence removes covariance terms and r equal variances add.

MATHEMATICAL STATEMENT

Each geometric wait has variance (1-p)/p squared; independence removes covariance terms and r equal variances add.

Learning goalExplain variance derivation for the negative binomial distribution.
Helpful foundationsBernoulli trials · binomial distribution · geometric distribution
Common trapNegative binomial fixes the required success count, not the number of trials.
Next connectionUse the PMF and moments in applied waiting-time problems