# Statistics Gamma Distribution – lesscss

By | October 18, 2019

# Statistics – Gamma Distribution

The gamma distribution represents continuous probability
distributions of two-parameter family. Gamma distributions are
devised with generally three kind of parameter combinations.

• A shape parameter \$ k \$ and a scale parameter \$ theta \$.

• A shape parameter \$ alpha = k \$ and an inverse scale
parameter \$ beta = frac{1}{ theta} \$, called as rate
parameter.

• A shape parameter \$ k \$ and a mean parameter \$ mu =
frac{k}{beta} \$.

Each parameter is a positive real numbers. The gamma distribution
is the maximum entropy probability distribution driven by
following criteria.

## Formula

\${E[X] = k theta = frac{alpha}{beta} gt 0 and is
fixed. \[7pt] E[ln(X)] = psi (k) + ln( theta) = psi(
alpha) – ln( beta) and is fixed. }\$

Where −

• \${X}\$ = Random variable.

• \${psi}\$ = digamma function.

## Characterization using shape \$ alpha \$ and rate \$ beta \$

### Probability density function

Probability density function of Gamma distribution is given as:

## Formula

\${ f(x; alpha, beta) = frac{beta^alpha x^{alpha – 1 } e^{-x
beta}}{Gamma(alpha)} where x ge 0 and alpha, beta
gt 0 }\$

Where −

• \${alpha}\$ = location parameter.

• \${beta}\$ = scale parameter.

• \${x}\$ = random variable.

### Cumulative distribution function

Cumulative distribution function of Gamma distribution is given
as:

## Formula

\${ F(x; alpha, beta) = int_0^x f(u; alpha, beta) du =
frac{gamma(alpha, beta x)}{Gamma(alpha)}}\$

Where −

• \${alpha}\$ = location parameter.

• \${beta}\$ = scale parameter.

• \${x}\$ = random variable.

• \${gamma(alpha, beta x)} \$ = lower incomplete gamma
function.

## Characterization using shape \$ k \$ and scale \$ theta \$

### Probability density function

Probability density function of Gamma distribution is given as:

## Formula

\${ f(x; k, theta) = frac{x^{k – 1 }
e^{-frac{x}{theta}}}{theta^k Gamma(k)} where x gt 0
and k, theta gt 0 }\$

Where −

• \${k}\$ = shape parameter.

• \${theta}\$ = scale parameter.

• \${x}\$ = random variable.

• \${Gamma(k)}\$ = gamma function evaluated at k.

### Cumulative distribution function

Cumulative distribution function of Gamma distribution is given
as:

## Formula

\${ F(x; k, theta) = int_0^x f(u; k, theta) du =
frac{gamma(k, frac{x}{theta})}{Gamma(k)}}\$

Where −

• \${k}\$ = shape parameter.

• \${theta}\$ = scale parameter.

• \${x}\$ = random variable.

• \${gamma(k, frac{x}{theta})} \$ = lower incomplete gamma
function.

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