Contents

- 1 Beta Distribution probabilities using R
- 2 Beta Type I Distribution
- 3 Beta Type I probabilities using dbeta() function in R
- 4 Numerical Problem for Beta Type I Distribution
- 5 Beta Type I cumulative probability using pbeta() function in R
- 6 Beta Type I Distribution Quantiles using qbeta() in R
- 7 Simulating Beta Type I random variable using rbeta() function in R
- 8 Endnote

## 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**

Binomial distribution in R

Poisson distribution in R

Geometric distribution in R

Negative Binomial distribution in R

Hypergeometric distribution in R

**Continuous Distributions Using R**

Uniform distribution in R

Exponential distribution in R

Log-Normal distribution in R

Normal distribution in R

Gamma distribution in R

Cauchy distribution in R

Laplace distribution in R

Logistic distribution in R

Weibull distribution in 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.

To learn more about R code for discrete and continuous probability distributions, please refer to the following tutorials:

Probability Distributions using R

Let me know in the comments below, if you have any questions on Beta Type I Distribution using R and your thought on this article.