# Beta Distribution probabilities using R

## Beta Distribution probabilities using R

In this tutorial, you will learn about how to use dbeta(), pbeta(), qbeta() and rbeta() functions in R programming language to compute the individual probabilities, cumulative probabilities, quantiles and to generate random sample for Beta Type I distribution.

Before we discuss R functions for Beta Type I distribution, let us see what is Beta Type I distribution.

## Beta Type I Distribution

Beta Type I distribution distribution is a continuous type probability distribution.

Let $X\sim \beta_1(\alpha,\beta)$. Then the probability distribution of $X$ is

 \begin{aligned} f(x)&= \begin{cases} \frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}, & 0\leq x\leq 1;\alpha,\beta>0 \\ 0, & Otherwise. \end{cases} \end{aligned}

where $\alpha$ is the shape parameter 1 and $\beta$ is the shape parameter 2 of Beta Type I distribution.

Read more about the theory and results of Beta Type I distribution here.

## Beta Type I probabilities using dbeta() function in R

For continuous probability distribution, density is the value of the probability density function at $x$ (i.e., $f(x)$).

The syntax to compute the probability density function for Beta Type I distribution using R is

dbeta(x,shape1, shape2)

where

• x : the value(s) of the variable and,
• shape1 : first parameter of beta distribution,
• shape2 : second parameter of beta distribution.

The dbeta() function gives the density for given value(s) x, shape1 and shape2.

## Numerical Problem for Beta Type I Distribution

To understand the four functions dbeta(), pbeta(), qbeta() and rbeta(), let us take the following numerical problem.

### Beta Type I Distribution Example

The daily proportion of major automobile accidents across the United States can be treated as a random variable having a beta distribution with $\alpha = 6$ and $\beta = 4$.

(a) Find the value of the density function at $x=0.35$.
(b) Plot the graph of Beta Type I probability distribution.
(c) Find the probability that, on a certain day, the percentage of major accidents is less than 80%.
(d) Find the probability that, on a certain day, the percentage of major accidents is more than 60%.
(e) Find the probability that, on a certain day, the percentage of major accidents is less than 80% but greater than 60%.
(f) Plot the graph of cumulative Beta Type I probabilities.
(g) What is the value of $c$, if $P(X\leq c) \geq 0.85$?
(h) Simulate 1000 Beta Type I distributed random variables with $\alpha= 6$ and $\beta = 4$.

Let $X$ denote the daily proportion of major automobile accidents across the United States. Given that $X\sim \beta_1(\alpha=6, \beta=4)$.

### Example 1: How to use dbeta() function in R?

To find the value of the density function at $x=0.35$ we need to use dbeta() function.

Let $X$ denote the daily proportion of major automobile accidents across the United States. Here $X\sim \beta_1(6,4)$.

First let us define the given parameters as

# parameter 1 shape1
alpha <- 6
# parameter 2 shape2
beta <- 4

The probability density function of $X$ is

 \begin{aligned} f(x)&= \frac{1}{B(6,4)}x^{6-1}(1-x)^{4-1},\\ &\quad\text{for } 0 \leq x \leq 1. \end{aligned}

For part (a), we need to find the density function at $x=0.35$. That is $f(0.35)$.

First I will show you how to calculate the value of the density function for given value of $x$. Then I will show you how to compute the same using dbeta() function in R.

(a) The value of the density function at $x=0.35$ is

 \begin{aligned} f(0.35)&= \frac{1}{B(6, 4)}*(0.35)^{6-1} (1-0.35)^{4-1}\\ &= \frac{\Gamma(10)}{\Gamma(6)\Gamma(4)}*(0.35)^{5} (0.65)^{3}\\ &=504*0.0052522 * 0.274625\\ &= 0.7269605 \end{aligned}

The above probability can be calculated using dbeta(0.35,6,4) function in R.

# Compute Beta Type I  probability
result1 <- dbeta(0.35,alpha,beta)
result1
[1] 0.7269605

### Example 2 Visualize Beta Type I probability distribution

Using dbeta() function we can compute Beta Type I distribution probabilities for given x, shape1 and shape2. To plot the probability density function of Beta Type I distribution, we need to create a sequence of x values and compute the corresponding probabilities.

# create a sequence of x values
x <- seq(0,1, by=0.02)
## Compute the Beta Type I pdf for each x
px<-dbeta(x,alpha,beta)

(b) Visualizing Beta Type I Distribution with dbeta() function and plot() function in R:

The probability density function of Beta Type I distribution with given 6 and 4 can be visualized using plot() function as follows:

## Plot the Beta Type I  probability dist
plot(x,px,type="l",xlim=c(0,1),ylim=c(0,max(px)),
lwd=3, col="darkred",ylab="f(x)")
title("PDF of Beta Type I  (alpha = 6, beta= 4)")

## Beta Type I cumulative probability using pbeta() function in R

The syntax to compute the cumulative probability distribution function (CDF) for Beta Type I distribution using R is

pbeta(q,shape1, shape2)

where

• q : the value(s) of the variable,
• shape1 : first parameter of beta distribution,
• shape2 : second parameter of beta distribution.

Using this function one can calculate the cumulative distribution function of Beta Type I distribution for given value(s) of q (value of the variable x), shape1 and shape2.

### Example 3: How to use pbeta() function in R?

In the above example, for part (c), we need to find the probability $P(X\leq 0.80)$.

(c) The probability that, on a certain day, the percentage of major accidents is less than 80% is

 \begin{aligned} P(X\leq 0.80) &=\int_0^{0.80} f(x)\; dx. \end{aligned}

## Compute cumulative Beta Type I  probability
result2 <- pbeta(0.80,alpha,beta)
result2
[1] 0.9143583

### Example 4: How to use pbeta() function in R?

In the above example, for part (d), we need to find the probability $P(X \geq 0.60)$.

To calculate the probability that a random variable $X$ is greater than a given number one can use the option lower.tail=FALSE in pbeta() function.

Above probability can be calculated easily using pbeta() function with argument lower.tail=FALSE as

$P(X \geq 0.60) =\int_{0.60}^1 f(x)\; dx$ = pbeta(0.60,alpha,beta,lower.tail=FALSE)

or by using complementary event as

$P(X \geq 0.60) = 1- P(X\leq 0.60)$= 1- pbeta(1,alpha,beta)

# compute cumulative Beta Type I  probabilities
# with lower.tail False
pbeta(0.60,alpha,beta,lower.tail=FALSE)
[1] 0.5173903

(d) The probability that, on a certain day, the percentage of major accidents is more than 60% is

 \begin{aligned} P(X\geq 0.60) &=\int_{0.60}^1 f(x)\; dx\\ &=0.5173903. \end{aligned}

# Using complementary event
1-pbeta(0.60,alpha,beta)
[1] 0.5173903

### Example 5: How to use pbeta() function in R?

One can also use pbeta() function to calculate the probability that the random variable $X$ is between two values.

(e) The probability that, on a certain day, the percentage of major accidents is less than 80% but greater than 60% can be written as $P(0.60 < X < 0.80)$.

 \begin{aligned} P(0.60 < X < 0.80) &= P(X< 0.80) -P(X < 0.60)\\ &= 0.9143583 - 0.4826097\\ &= 0.4317486 \end{aligned}

The above probability can be calculated using pbeta() function as follows:

result3 <- pbeta(0.80,alpha,beta)-pbeta(0.60,alpha,beta)
result3
[1] 0.4317486

### Example 6: Visualize the cumulative Beta Type I probability distribution

Using pbeta() function we can compute Beta Type I cumulative probabilities (CDF) for given x, shape1 and shape2. To plot the CDF of Beta Type I distribution, we need to create a sequence of x values and compute the corresponding cumulative probabilities.

# create a sequence of x values
x <- seq(0,1, by=0.02)
## Compute the Beta Type I pdf for each x
Fx <- pbeta(x,alpha,beta)

(f) Visualizing Beta Type I Distribution with pbeta() function and plot() function in R:

The cumulative probability distribution of Beta Type I distribution with given x, shape1 and shape2 can be visualized using plot() function as follows:

## Plot the Beta Type I  probability dist
plot(x,Fx,type="l",xlim=c(0,1),ylim=c(0,1),
lwd=3, col="darkred",ylab="f(x)")
title("Distribution Function of Beta Type I  (alpha = 6, beta= 4)")

## Beta Type I Distribution Quantiles using qbeta() in R

The syntax to compute the quantiles of Beta Type I distribution using R is

qbeta(p,shape1,shape2)

where

• p : the value(s) of the probabilities,
• shape1 : first parameter of beta distribution,
• shape2 : second parameter of beta distribution.

The function qbeta(p,shape1,shape2) gives $100*p^{th}$ quantile of Beta Type I distribution for given value of p, shape1 and shape2.

The $p^{th}$ quantile is the smallest value of Beta Type I random variable $X$ such that $P(X\leq x) \geq p$.

It is the inverse of pbeta() function. That is, inverse cumulative probability distribution function for Beta Type I distribution.

### Example 7: How to use qbeta() function in R?

In part (g), we need to find the value of $c$ such a that $P(X\leq c) \geq 0.85$. That is we need to find the $85^{th}$ quantile of given Beta Type I distribution.

alpha <- 6
beta <- 4
prob <- 0.85
# compute the quantile for Beta Type I  dist
qbeta(0.85,alpha, beta)
[1] 0.7586159

The $85^{th}$ percentile of given Beta Type I distribution is 0.7586159.

### Visualize the quantiles of Beta Distribution

The quantiles of Beta distribution with given p, shape1 and shape2 can be visualized using plot() function as follows:

p <- seq(0,1,by=0.02)
qx <- qbeta(p,alpha,beta)
# Plot the quantiles of Beta Type I  dist
plot(p,qx,type="l",lwd=2,col="darkred",
ylab="quantiles",
main="Quantiles of Beta(alpha= 6,beta = 4)")

## Simulating Beta Type I random variable using rbeta() function in R

The general R function to generate random numbers from Beta Type I distribution is

rbeta(n,shape1,shape2)

where,

• n : the sample observations,
• shape1 : first parameter of beta distribution,
• shape2 : second parameter of beta distribution.

The function rbeta(n,shape1,shape2) generates n random numbers from Beta Type I distribution with given shape1 and shape2.

### Example 8: How to use rbeta() function in R?

In part (h), we need to generate 1000 random numbers from Beta Type I distribution with given $shape1 = 6$ and $shape2=4$.

(h) We can use rbeta(1000,alpha,beta) function to generate random numbers from Beta Type I distribution.

## initialize sample size to generate
n <- 1000
# Simulate 1000 values From Beta Type I  dist
x_sim <- rbeta(n,alpha,beta)

The below graphs shows the density of the simulated random variables from Beta Type I Distribution.

## Plot the simulated data
plot(density(x_sim),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main="Simulated data from Beta(6,4) dist")

If you use same function again, R will generate another set of random numbers from $\beta_1(6,4)$.

# Simulate 1000 values From Beta Type I  dist
x_sim_2 <- rbeta(n,alpha,beta)
## Plot the simulated data
plot(density(x_sim_2),xlab="Simulated x",ylab="density",
lwd=5,col="blue",
main="Simulated data from Beta(6,4) dist")

For the simulation purpose to reproduce same set of random numbers, one can use set.seed() function.

# set seed for reproducibility
set.seed(1457)
# Simulate 1000 values From Beta Type I  dist
x_sim_3 <- rbeta(n,alpha,beta)
## Plot the simulated data
plot(density(x_sim_3),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main="Simulated data from Beta(6,4) dist")
set.seed(1457)
# Simulate 1000 values From Beta Type I  dist
x_sim_4 <- rbeta(n,alpha,beta)
## Plot the simulated data
plot(density(x_sim_4),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main="Simulated data from Beta(6,4) dist")

Since we have used set.seed(1457) function, R will generate the same set of Beta Type I distributed random numbers.

To learn more about other discrete and continuous probability distributions using R, go through the following tutorials:

Discrete Distributions Using R

Continuous Distributions Using R

## Endnote

In this tutorial, you learned about how to compute the probabilities, cumulative probabilities and quantiles of Beta Type I distribution in R programming. You also learned about how to simulate a Beta Type I distribution using R programming.