Discrete Distribution Examples

Worked examples for common discrete distributions. For each example the range, PMF, and expected moment values are given. All results were verified against known analytical formulae.

Bernoulli Distribution 

The Bernoulli distribution models a single binary trial with success probability \(p\).

Parameters: \(p = 0.6\)

Support: 0, 1

Range:  0,1
PMF:    0.6 if x == 1 else 0.4

Known moments about origin:

Order	Value
mu_1	0.6 (= p)
mu_2	0.6 (= p, since X^2 = X for Bernoulli)
mu_3	0.6
mu_4	0.6

Known central moments:

Order	Value
mu_1	0 (always)
mu_2	0.24 (= p*(1-p))

Binomial Distribution 

The Binomial distribution models the number of successes in \(n\) independent Bernoulli trials.

Parameters: \(n = 10\), \(p = 0.4\)

Support: 0, 1, 2, …, 10

Range:  0,1,2,3,4,5,6,7,8,9,10
PMF:    (factorial(10) / (factorial(x) * factorial(10-x))) * 0.4**x * 0.6**(10-x)

Known analytical values:

Mean      = n*p        = 4.0
Variance  = n*p*(1-p)  = 2.4
Std Dev   = sqrt(2.4)  ≈ 1.5492
Skewness  = (1-2p)/sqrt(n*p*(1-p)) ≈ 0.1291
Kurtosis  = 3 + (1-6p(1-p))/(n*p*(1-p)) ≈ 3.0583

Geometric Distribution (failures before first success)

The Geometric distribution models the number of failures before the first success, with support starting at 0.

Parameters: \(p = 0.3\)

Support: 0, 1, 2, 3, …

Range:  0,1,2,3,...
PMF:    0.3 * (0.7 ** x) if x >= 0 else 0

Known analytical values:

Mean      = (1-p)/p     = 7/3  ≈ 2.3333
Variance  = (1-p)/p²    = 70/9 ≈ 7.7778
Skewness  = (2-p)/sqrt(1-p) ≈ 2.031
Kurtosis  = 9 + p²/(1-p)   ≈ 9.129

Geometric Distribution (trials until first success)

Alternative parameterisation: support starts at 1.

Parameters: \(p = 0.3\)

Range:  1,2,3,4,...
PMF:    0.3 * (0.7 ** (x-1)) if x >= 1 else 0

Known analytical values:

Mean      = 1/p         = 10/3 ≈ 3.3333
Variance  = (1-p)/p²    ≈ 7.7778  (same as above)

Poisson Distribution 

The Poisson distribution models the number of events in a fixed interval when events occur at constant rate \(\lambda\).

Parameters: \(\lambda = 3\)

Support: 0, 1, 2, 3, …

Range:  0,1,2,3,...
PMF:    (exp(-3) * 3**x) / factorial(x) if x >= 0 else 0

Known analytical values:

Mean      = λ           = 3.0
Variance  = λ           = 3.0
Skewness  = 1/sqrt(λ)   ≈ 0.5774
Kurtosis  = 3 + 1/λ     = 3.3333

Tip

The Poisson distribution’s PMF decays very quickly for large \(x\) relative to \(\lambda\). With \(\lambda = 3\), 200 terms captures more than \(10^{-200}\) of the total probability.

Negative Binomial Distribution 

The Negative Binomial models the number of failures before the \(r\)-th success.

Parameters: \(r = 2\), \(p = 0.5\)

Support: 0, 1, 2, 3, …

Range:  0,1,2,3,...
PMF:    (factorial(x+1) / (factorial(x) * factorial(1))) * 0.5**2 * 0.5**x if x >= 0 else 0

Known analytical values:

Mean      = r*(1-p)/p   = 2
Variance  = r*(1-p)/p²  = 4

Uniform Discrete Distribution 

Equal probability over a finite set of values.

Parameters: values 1 through 6 (fair die)

Range:  1,2,3,4,5,6
PMF:    1/6

Known analytical values:

Mean      = (n+1)/2     = 3.5
Variance  = (n²-1)/12   = 35/12 ≈ 2.9167
Skewness  = 0           (symmetric)
Kurtosis  = 3 - (6(n²+1))/(5(n²-1)) ≈ 2.5143

Tips for All Discrete Distributions 

Set the moment reference to Mean (a = μ) and order ≥ 4 to see skewness and kurtosis in the Statistics tab.
Cross-check the first moment (μ₁) against the known analytical mean before trusting higher-order results.
For the Poisson distribution with small \(\lambda\), increasing Calc precision to 10 helps accuracy for the 4th moment and above.
If validation fails with sum ≠ 1, verify the PMF guard (if x >= 0 else 0) and check for typos in the exponent.