# R - Weibull Distribution

Weibull Distribution is a continuous probability distribution and it is widely used analyze life data, model failure times and access product reliability. It can also fit a huge range of data from many other fields like economics, hydrology, biology, engineering sciences.

The probability density function (pdf) of weibull distribution is defined as: Where, k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution.

The cumulative distribution function (cdf) evaluated at x, is the probability that the random variable (X) will take a value less than or equal to x. The cdf of weibull distribution is defined as: In R, there are four functions which can be used to generate weibull distribution.

### Syntax

```dweibull(x, shape, scale)
pweibull(q, shape, scale)
qweibull(p, shape, scale)
rweibull(n, shape, scale)
```

### Parameters

 `x` `Required. `Specify a vector of numbers. `q` `Required. `Specify a vector of numbers. `p` `Required. `Specify a vector of probabilities. `n` `Required. `Specify number of observation (sample size). `shape` `Optional. `Specify shape parameter of the distribution. Must be a positive number. `scale` `Optional. `Specify scale parameter of the distribution. Must be a positive number. Default is 1.

## dweibull()

The dweibull() function measures probability density function (pdf) of the distribution.

```#creating a sequence of values between
#-1 to 2 with a difference of 0.01
x <- seq(-1, 2, by=0.01)

y <- dweibull(x, 2, 1)

#naming the file
png(file = "weibull.png")

#plotting the graph
plot(x, y, col="blue")

#saving the file
dev.off()
```

The output of the above code will be: ## pweibull()

The pweibull() function returns cumulative distribution function (cdf) of the distribution.

```#creating a sequence of values between
#-1 to 3 with a difference of 0.01
x <- seq(-1, 3, by=0.01)

y <- pweibull(x, 2, 1)

#naming the file
png(file = "weibull.png")

#plotting the graph
plot(x, y, col="blue")

#saving the file
dev.off()
```

The output of the above code will be: ## qweibull()

The qweibull() function takes the probability value and returns cumulative value corresponding to probability value of the distribution.

```#creating a sequence of probability from
#0 to 1 with a difference of 0.01
x <- seq(0, 1, by=0.01)

y <- qweibull(x, 2, 1)

#naming the file
png(file = "weibull.png")

#plotting the graph
plot(x, y, col="blue")

#saving the file
dev.off()
```

The output of the above code will be: ## rweibull()

The rweibull() function generates a vector containing specified number of random numbers from the specified weibull distribution. In the example below, a histogram is plotted to visualize the result.

```#fixing the seed to maintain the
#reproducibility of the result
set.seed(10)
x <- 10000

#creating a vector containing 10000 random
#values from the given weibull distribution
y <- rweibull(x, 2, 1)

#naming the file
png(file = "weibull.png")

#plotting the graph
hist(y, col="blue")

#saving the file
dev.off()
```

The output of the above code will be: 5