Discrete Distribution Calculator

Distribution

Select the model that matches your discrete outcome.

Output

PMF is exact probability at k. CDF is P(X ≤ k).

k (point)

Used for PMF/CDF badges. Table uses k-range.

k from

k to

Parameters

Only the selected distribution parameters are used.

Binomial n (trials)

p (success probability)

Poisson λ (rate)

Geometric support

Negative binomial r (successes)

k meaning

Population N

Success states K

Draws n

Uniform a (min)

Uniform b (max)

Example data table

Example: Binomial with n = 10 and p = 0.50, showing k = 0..6.

k	PMF	CDF
0	0.0009765625	0.0009765625
1	0.0097656250	0.0107421875
2	0.0439453125	0.0546875000
3	0.1171875000	0.1718750000
4	0.2050781250	0.3769531250
5	0.2460937500	0.6230468750
6	0.2050781250	0.8281250000

Formulas used

This tool uses stable log-factorials for combinations: log(n!) = lgamma(n+1).

Binomial

P(X=k)=C(n,k)p^k(1-p)^(n-k)

Mean = np, Variance = np(1−p).

Poisson

P(X=k)=e^(−λ) λ^k / k!

Mean = λ, Variance = λ.

Geometric

P(X=k)=(1−p)^k p (k=0,1,2,...)

Alternate support: P(X=k)=(1−p)^(k−1)p for k≥1.

Negative binomial

P(X=k)=C(k+r−1,k)(1−p)^k p^r

Here k counts failures before r-th success.

Hypergeometric

P(X=k)= C(K,k) C(N−K, n−k) / C(N,n)

Sampling is without replacement.

Discrete uniform

P(X=k)=1/(b−a+1) for a≤k≤b

Mean=(a+b)/2, Variance=((b−a+1)^2−1)/12.

How to use this calculator

Select the discrete distribution that fits your scenario.
Enter the required parameters and your target k.
Set a k from and k to range for tables.
Press Calculate to display results above the form.
Use the download buttons to export the latest computed table.

FAQs

1) What is a discrete distribution?

It models outcomes that are countable, like counts, trials, or selected items. Probabilities are assigned to integer values, and the PMF sums to one across its support.

2) What is the difference between PMF and CDF?

PMF gives the probability at one exact value, P(X=k). CDF gives the cumulative probability up to that value, P(X≤k), which helps answer “at most” questions.

3) How do I choose between binomial and Poisson?

Use binomial for a fixed number of independent trials with the same success chance. Use Poisson for event counts over time or space with an average rate λ.

4) Why does the hypergeometric distribution matter?

Hypergeometric is for sampling without replacement, so probabilities change after each draw. It is common in quality checks, audits, and card draws from a deck.

5) Why are combinations computed with log-factorials?

Direct factorials overflow quickly for large inputs. Log-factorials using lgamma keep calculations stable, then exponentiation returns usable probabilities for many practical sizes.

6) My table range probability is less than one. Is that wrong?

Not necessarily. It means your chosen k-range does not cover all support values. Expand the range to capture more mass, especially for Poisson or negative binomial tails.

7) How can I export and share results?

After calculating, use the CSV button for spreadsheet analysis. Use the PDF button for a lightweight printable summary, including the first rows of the table.