Negative Binomial Distribution

The Probabilistic Experiment Behind negative binomial distribution

Notation Used

Parameters

Probability Mass Function (PMF) and Support (Range)

Cumulative Distribution Function (CDF)

Expected Value (Mean)

Variance and Standard Deviation

Applications and Examples

Negative Binomial Distribution: Trials Until the r-th Success

The negative binomial distribution extends the geometric experiment by continuing the sequence of independent Bernoulli trials until a specified number of successes is reached. Rather than stopping at the first success, the experiment proceeds until the

r

-th success occurs. The random variable measures the total number of trials required to reach this target, capturing variability in how long repeated success takes to accumulate.

The Probabilistic Experiment Behind negative binomial distribution

The negative binomial distribution generalizes the geometric distribution by counting the number of trials required until a fixed number of successes is reached, rather than just the first success. Trials are independent Bernoulli experiments with constant success probability, and the process continues until the target number of successes is achieved.

Here, the random variable counts the total number of trials, including both successes and failures. The number of successes is fixed in advance, while the number of failures — and thus the total length of the experiment — is random.

This distribution is useful when success must occur multiple times before stopping, and the timing of those successes is uncertain. When the required number of successes is

1

, the negative binomial distribution reduces exactly to the geometric distribution.

Learn More About Distinguishing Discrete Distributions →

Example:

Flipping a coin until you obtain

3

heads. If

X=7

, this means the third head appears on the seventh flip. The sequence ends at the moment the third success occurs.

See More Examples of Negative Binomial Distribution →

Notation Used

X \sim \text{NegBin}(r, p)

X \sim \text{NB}(r, p)

— distribution of the random variable.

\text{NegativeBinomial}(r, p)

— used to denote the distribution itself (not the random variable).

\text{NB}(r, p)

— common shorthand, especially in statistical software.

P(X = k) = \binom{k - 1}{r - 1} p^r (1 - p)^{k - r}, \quad \text{for } k = r, r+1, r+2, \ldots

— probability mass function (PMF) (trials until

r

-th success), where:

r

— number of successes desired

p

— probability of success on each trial

k

— total number of trials until

r

-th success

\binom{k-1}{r-1} = \frac{(k-1)!}{(r-1)!(k-r)!}

— binomial coefficient

Alternative PMF formulation:

P(X = k) = \binom{k + r - 1}{k} p^r (1 - p)^k, \quad \text{for } k = 0, 1, 2, \ldots

— number of failures before

r

-th success

Alternative notations:

q = 1 - p

— probability of failure, so PMF can be written as

P(X = k) = \binom{k-1}{r-1} p^r q^{k-r}

Key properties:

E[X] = \frac{r}{p}

— expected value (mean)

\text{Var}(X) = \frac{r(1-p)}{p^2}

— variance

Relationship to geometric distribution:

\text{NegBin}(1, p) = \text{Geom}(p)

— negative binomial is a generalization of geometric distribution

See All Probability Symbols and Notations →

Parameters

𝑟

: number of successes to achieve (a positive integer)

𝑝

: probability of success in each trial, with

0<𝑝≤1

This distribution models the number of trials needed to observe

𝑟

successes, assuming each trial is independent and has the same probability

𝑝

of success.

The outcomes are integers

𝑟

𝑟+1

𝑟+2

,…, since at least

𝑟

trials are needed.

𝑟

controls the target (how many successes), and

𝑝

controls the chance of achieving each one — together, they define how spread out or concentrated the distribution is.

Probability Mass Function (PMF) and Support (Range)

The probability mass function (PMF) of a negative binomial distribution is given by:

P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}, \quad k = r, r+1, r+2, \ldots

where

\binom{k-1}{r-1} = \frac{(k-1)!}{(r-1)!(k-r)!}

is the binomial coefficient.

Fixed Number of Successes: The negative binomial distribution models the number of trials needed to achieve exactly

r

successes in a sequence of independent Bernoulli trials.

Support (Range of the Random Variable):
* The random variable

X

can take on values

r, r+1, r+2, \ldots

(integers starting from

r

).
*

X = k

means the

r

-th success occurs on the

k

-th trial.
* The support is thus a countably infinite set.

Logic Behind the Formula:
*

\binom{k-1}{r-1}

: The number of ways to arrange

r-1

successes in the first

k-1

trials (the

k

-th trial must be the

r

-th success)
*

p^r

: The probability of getting exactly

r

successes
*

(1-p)^{k-r}

: The probability of getting exactly

k-r

failures
* The total probability sums to 1:

\sum_{k=r}^{\infty} P(X = k) = \sum_{k=r}^{\infty} \binom{k-1}{r-1} p^r (1-p)^{k-r} = 1

* This follows from the negative binomial series expansion.

Negative Binomial Distribution

Trials until r-th success (generalization of geometric)

Number of Successes (r): 3

Success Probability (p): 0.40

Explanation

The negative binomial distribution models the number of trials needed to achieve $r$ successes in repeated independent trials. The probability mass function is $P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}$ . The expected value is $E[X] = \frac{r}{p}$ and the variance is $\text{Var}(X) = \frac{r(1-p)}{p^2}$ . This distribution is used for modeling scenarios like the number of calls until $r$ sales are made, games played until $r$ wins are achieved, or attempts until $r$ successes occur.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a negative binomial distribution is given by:

F_X(k) = P(X \leq k) = \sum_{i=r}^{k} \binom{i-1}{r-1} p^r (1-p)^{i-r}

Where:

r

= number of successes desired (fixed, positive integer)

p

= probability of success on each trial

k

= number of trials until the

r

-th success (where

k \geq r

)

\binom{i-1}{r-1}

= binomial coefficient

Intuition Behind the Formula

Definition: The CDF gives the probability that the

r

-th success occurs on or before trial

k

.

Summation of Probabilities:
We sum the PMF values from the minimum possible value (

r

trials) up to

k

trials:

P(X \leq k) = P(X=r) + P(X=r+1) + P(X=r+2) + \cdots + P(X=k)

Alternative Formulation via Regularized Incomplete Beta Function:
The CDF can also be expressed using the regularized incomplete beta function:

F_X(k) = I_p(r, k-r+1)

This relationship connects the negative binomial distribution to the beta distribution and is often used in statistical software for efficient computation.

Complementary Form:
The probability that the

r

-th success occurs after trial

k

is:

P(X > k) = 1 - F_X(k)

Negative Binomial Distribution CDF

CDF for waiting time until r-th success

Number of Successes (r): 3

Success Probability (p): 0.40

CDF Explanation

The negative binomial CDF is

F(k) = P(X \leq k) = \sum_{i=r}^{k} \binom{i-1}{r-1} p^r (1-p)^{i-r}

for

k \geq r

. This gives the probability that the

r

-th success occurs on or before trial

k

. The CDF begins at

k = r

(minimum trials needed for

r

successes) with value

F(r) = p^r

. As

k

increases, the CDF approaches 1.0. The distribution generalizes the geometric distribution (which is the special case

r = 1

), and its shape depends on both the number of required successes

r

and the success probability

p

Expected Value (Mean)

As explained in the general case for calculating expected value, the expected value of a discrete random variable is computed as a weighted sum where each possible value is multiplied by its probability:

E[X] = \sum_{x} x \cdot P(X = x)

For the negative binomial distribution, we apply this general formula to the specific probability mass function of this distribution.

Formula

E[X] = \frac{r}{p}

Where:

r

= number of successes desired (fixed, positive integer)

p

= probability of success on each trial

Derivation and Intuition

The negative binomial random variable

X

represents the number of trials needed to achieve

r

successes. It can be viewed as the sum of

r

independent geometric random variables, where each represents the number of trials needed to achieve one additional success.

Since each geometric variable has expected value

\frac{1}{p}

, and we need

r

such successes:

E[X] = r \cdot \frac{1}{p} = \frac{r}{p}

This result follows directly from the linearity of expectation applied to the sum of

r

geometric random variables.

The result

E[X] = \frac{r}{p}

extends the geometric distribution's intuition: if you need one success and expect

\frac{1}{p}

trials, then needing

r

successes should require

r

times as many trials on average.

Example

Consider rolling a die until you get three 6's, where

r = 3

and

p = \frac{1}{6}

E[X] = \frac{3}{1/6} = 18

On average, you expect to roll the die 18 times before accumulating three 6's. This is exactly three times the expected wait for a single 6.

Variance and Standard Deviation

The variance of a discrete random variable measures how spread out the values are around the expected value. It is computed as:

\mathrm{Var}(X) = \mathbb{E}[(X - \mu)^2] = \sum_{x} (x - \mu)^2 P(X = x)

Or using the shortcut formula:

\mathrm{Var}(X) = \mathbb{E}[X^2] - \mu^2

For the negative binomial distribution, we apply this formula to derive the variance.

Formula

\mathrm{Var}(X) = \frac{r(1-p)}{p^2}

Where:

r

= number of successes desired (fixed, positive integer)

p

= probability of success on each trial

(1-p) = q

= probability of failure on each trial

Derivation and Intuition

The negative binomial random variable can be viewed as the sum of

r

independent geometric random variables, each representing the trials needed for one additional success.

Since each geometric variable has variance

\frac{1-p}{p^2}

, and variances add for independent variables:

\mathrm{Var}(X) = r \cdot \frac{1-p}{p^2} = \frac{r(1-p)}{p^2}

The result

\mathrm{Var}(X) = \frac{r(1-p)}{p^2}

extends the geometric distribution's variance by a factor of

r

. As with the geometric case, variance increases rapidly as

p

decreases (rare successes create high variability) and grows linearly with the number of required successes

r

Standard Deviation

\sigma = \sqrt{\frac{r(1-p)}{p^2}} = \frac{\sqrt{r(1-p)}}{p}

Example

Consider rolling a die until you get three 6's, where

r = 3

and

p = \frac{1}{6}

\mathrm{Var}(X) = \frac{3 \times \frac{5}{6}}{(\frac{1}{6})^2} = \frac{\frac{15}{6}}{\frac{1}{36}} = \frac{5}{2} \times 36 = 90

\sigma = \sqrt{90} \approx 9.487

The variance of 90 and standard deviation of about 9.5 indicate high variability around the expected 18 rolls. The actual number of rolls needed could vary substantially from this average.

Applications and Examples

### Practical Example

Suppose you're flipping a coin until you get

r = 3

heads, where the probability of heads is

p = 0.5

. The probability that you need exactly

k = 6

flips to get your third head is:

P(X = 6) = \binom{6-1}{3-1} (0.5)^3 (0.5)^{6-3} = \binom{5}{2} (0.5)^3 (0.5)^3 = 10 \cdot 0.125 \cdot 0.125 = 0.15625

This means there's a 15.625% chance that you'll need exactly 6 flips to get your third head.

Note: The geometric distribution is a special case of the negative binomial distribution where

r = 1