Gamma Distribution Calculator

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.

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

Calculator

Gamma Distribution 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)

How to use Gamma Distribution Calculator?

Step 1 - Enter the location parameter (alpha)

Step 2 - Enter the Scale parameter (beta)

Step 3 - Enter the Value of x

Step 4 - Click on “Calculate” button to calculate gamma distribution probabilities

Step 5 - Calculate Probability Density

Step 6 - Calculate Probability X less x

Step 7 - Calculate Probability X greater than x

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 and variance of gamma distribution $G(\alpha,\beta)$ are $\mu_1^\prime =\alpha\beta$ and $\mu_2 =\alpha\beta^2$ respectively.

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 numerical problem solved using Gamma Distribution Calculator with step by step procedure to calculate probabilities.

Gamma Distribution Calculator Example 1

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

Gamma Probability Calculator Example 2

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

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

Gamma Distribution Calculator Example 3

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

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

Gamma Distribution Calculator Example 4

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 like Gamma Distribution Calculator. Click on Theory to read more about Gamma distribution,graph of gamma distribution,M.G.F and C.G.F of gamma distribution.

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