Gamma distribution is used to model a continuous random variable which takes positive values. Gamma distribution is widely used in science and engineering to model a skewed distribution.

A gamma distrubution calculator can be used to calculate the probability density function (PDF), cumulative distribution function (CDF) and random numbers from a gamma distribution.

Use Gamma Distribution Calculator to calculate the probability density and lower and upper cumulative probabilities for Gamma distribution with parameter $\alpha$ and $\beta$.

## How to use Gamma Distribution Calculator with steps by steps Procedure?

• Enter the location parameter (alpha)
• Enter the Scale parameter (beta)
• Enter the Value of x
• Click on “Calculate” button to calculate gamma distribution probabilities
• Calculate Probability Density
• Calculate Probability X less x
• Calculate Probability X greater than x

## Gamma Distribution Probability Calculator

Gamma Probability Calculator
Location Parameter $\alpha$:
Scale Parameter $\beta$
Value of x
Gamma Probability Results
Probability density : f(x)
Probability X less than x: P(X < x)
Probability X greater than x: P(X > x)

Here’s the brief explanation about each fields in gamma distribution calculator.

• Location Parameter (alpha): This field represents the shape parameter (α) of the Gamma distribution. It determines the shape of the distribution curve. A higher α value makes the curve more peaked, while a lower value makes it more spread out.

• Scale Parameter (beta): The scale parameter (β) determines the scale of the Gamma distribution. It affects the spread of the curve. A larger β value results in a wider curve, while a smaller value narrows it.

• Value of x: This field represents the specific value of the random variable (X) for which you want to calculate probabilities or density. You input the value of X here to find the probabilities associated with it.

• Calculate Button: After entering the α, β, and x values, you click this button to initiate the calculations. It computes the probability density function (PDF), cumulative distribution function (CDF), and other related probabilities.

• Probability Density (f(x)): This field displays the probability density at the specified value of X. In other words, it shows how likely it is for the random variable to take on the given value.

• Probability X less than x (P(X < x)): This field provides the probability that the random variable X is less than the specified value x. It quantifies the likelihood that X falls below a certain threshold.

• Probability X greater than x (P(X > x)): This field gives the probability that the random variable X is greater than the specified value x. It represents the likelihood of X exceeding a particular threshold.

## Gamma Distribution Definition

A continuous random variable $X$ is said to have an gamma distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by

\begin{align*} f(x)&= \begin{cases} \frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha -1}e^{-x/\beta}, & x>0;\alpha, \beta >0; \\ 0, & Otherwise. \end{cases} \end{align*}

In notation, it can be written as $X\sim G(\alpha, \beta)$.

## Another form of gamma distribution is

\begin{align*} f(x)&= \begin{cases} \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha -1}e^{-\beta x}, & x>0;\alpha, \beta >0 \\ 0, & Otherwise. \end{cases} \end{align*}

## Mean and Variance of Gamma Distribution

The mean of gamma distribution $G(\alpha,\beta)$ is $\mu_1^\prime =\alpha\beta$ and and variance of gamma distribution is $\mu_2 =\alpha\beta^2$

The probabilities can be computed using MS EXcel or R function pgamma(). The percentiles or quantiles can be computed using MS EXcel or R function qgamma(). The probabilities can also be computed using incomplete gamma functions.

Read below gamma distribution examples solutions using Gamma Distribution Calculator with steps by steps procedure to calculate probabilities.

## Probability Calculation for a Gamma Distribution

Suppose that $Y$ has the gamma distribution with parameter $\alpha$ (shape) =10 and $\beta$ (scale)=2.

Use R to calculate the

a. probability that $Y$ is between 2 and 8, b. $90^{th}$ percentile of gamma distribution.

### Solution

Given that $X\sim G(10,2)$ distribution. That is $\alpha= 10$ and $\beta=2$.

The probability density function (pdf) of gamma distribution $X$ is

\begin{aligned} f(x;\alpha,\beta)&= \frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha -1}e^{-\frac{x}{\beta}}, x>0;\alpha, \beta >0 \\ &= \frac{1}{2^{10} \Gamma(10)} x^{10 -1}e^{-\frac{x}{2}}, x>0 \end{aligned}

a. The probability that $2 < X < 8$ is

\begin{aligned} P(2 < X < 8) &= P(X < 8) - P(X < 2)\\ &=\int_0^{8}f(x)\; dx - \int_0^{2}f(x)\; dx\\ &= 0.0081 -0\\ &=0.0081 \end{aligned}

b. Let the $90^{th}$ percentile be $Q$.

\begin{aligned} & P(X < Q) = 0.9\\ \Rightarrow &\int_0^{Q}f(x)\; dx=0.9\\ \Rightarrow &Q= 28.412 \end{aligned}

Thus $90^{th}$ percentile of the given gamma distribution is 28.412.

## Finding Probability in a Gamma Distribution

If a random variable $X$ has a gamma distribution with $\alpha=4.0$ and $\beta=3.0$, find $P(5.3 < X < 10.2)$.

### Solution

Given that $X\sim G(4,3)$ distribution. That is $\alpha= 4$ and $\beta=3$.

The probability density function (pdf) of gamma distribution $X$ is

\begin{aligned} f(x;\alpha,\beta)&= \frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha -1}e^{-\frac{x}{\beta}}, x>0;\alpha, \beta >0 \\ &= \frac{1}{3^{4} \Gamma(4)} x^{4 -1}e^{-\frac{x}{3}}, x>0 \end{aligned}

The probability that $5.3 < X < 10.2$ is

\begin{aligned} P(5.3 < X < 10.2) &= P(X < 10.2) - P(X < 5.3)\\ &=\int_0^{10.2}f(x)\; dx - \int_0^{5.3}f(x)\; dx\\ &= 0.4416 -0.1034\\ &=0.3382 \end{aligned}

## Analyzing Probabilities in a Standard Gamma Distribution

Let $X$ have a standard gamma distribution with $\alpha=3$. Find

a. $P(2\leq X \leq 6)$ b. $P(X>8)$ c. $P(X\leq 6)$

### Solution

Given that $X\sim G(3,1)$ distribution, which is a standard gamma distribution. That is $\alpha= 3$ and $\beta=1$.

The probability density function (pdf) of gamma distribution $X$ is

\begin{aligned} f(x;\alpha,\beta)&= \frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha -1}e^{-\frac{x}{\beta}}, x>0;\alpha, \beta >0 \\ &= \frac{1}{1^{3} \Gamma(3)} x^{3 -1}e^{-\frac{x}{1}}, x>0 \end{aligned}

a. The probability that $2 < X < 6$ is

\begin{aligned} P(2 < X < 6) &= P(X < 6) - P(X < 2)\\ &=\int_0^{6}f(x)\; dx-\int_0^{2}f(x)\; dx\\ &= 0.938 -0.3233\\ &=0.6147 \end{aligned}

b. The probability that $X > 8$ is

\begin{aligned} P(X > 8) &= 1- P(X \leq 8)\\ &=1- \int_0^{8}f(x)\; dx\\ &= 1-0.9862\\ &=0.0138 \end{aligned}

c. The probability that $X \leq 6$ is

\begin{aligned} P(X \leq 6)&= \int_{0}^{6} f(x)\; dx\\ &=0.938 \end{aligned}

## Mean and Variance Calculation for a Gamma Distribution

Time spend on the internet follows a gamma distribution is a gamma distribution with mean 24 $min$ and variance 78 $min^2$.

Find the

a. parameters of gamma distribution, c. probability that time spend on the internet is between 22 to 38 minutes, b. probability that time spend on the internet is less than 28 minutes.

### Solution

Let $X$ be the time spend on the internet. Given that $X\sim G(\alpha, \beta)$. The mean of $G(\alpha,\beta)$ distribution is $\alpha\beta$ and the variance is $\alpha\beta^2$.

Given that $mean =\alpha\beta=24$ and $V(X)=\alpha\beta^2=78$.

a. Thus $\beta=\frac{78}{24}=3.25$ and $\alpha = 24/3.25= 7.38$ (rounded to two decimal)

The probability density function (pdf) of gamma distribution $X$ is

\begin{aligned} f(x;\alpha,\beta)&= \frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha -1}e^{-\frac{x}{\beta}}, x>0;\alpha, \beta >0 \\ &= \frac{1}{3.25^{7.38} \Gamma(7.38)} x^{7.38 -1}e^{-\frac{x}{3.25}}, x>0 \end{aligned}

b. The probability that $22 < X < 38$ is

\begin{aligned} P(22 < X < 38) &= P(X < 38) - P(X < 22)\\ &=\int_0^{38}f(x)\; dx-\int_0^{22}f(x)\; dx\\ &= 0.9295 -0.4572\\ &=0.4722 \end{aligned}

b. The probability that $X < 28$ is

\begin{aligned} P(X < 28) &=\int_0^{28}f(x)\; dx\\ &= 0.7099 \end{aligned}

## Conclusion

I hope you find above article on Gamma Distribution Calculator educational. We have covered gamma calculator and gamma distribution examples and solutions step by step.Click on Theory to read more about Gamma distribution,graph of gamma distribution,M.G.F and C.G.F of gamma distribution.