Note that you might have to install tidyverse package if it is not already installed. You can installed it by writing install.package(tidyverse) in your console.

library(tidyverse)

In our handout we worked with random variable \(X\) such that \(X \sim Exp(0.1)\) which indicated that \(\lambda = 0.1\)

Also remember that probability density function (pdf) for the Exponential Distribution is

\(f(x) = \lambda e^{-\lambda x}\) for \(x\geq0\)

We had manually tried to plot the pdf by calculating \(f(x)\) for some values of \(x\). For instance

when \(x = 0\), then \(f(x) = 0.1e^{-0.1\times0} = 0.1\)
when \(x = 1\), then \(f(x) = 0.1e^{-0.1\times1} \approx 0.09\)
when \(x = 2\), then \(f(x) = 0.1e^{-0.1\times2} \approx 0.08\)

Note that X is continous so we could calculate \(f(1.5)\) or any other positive decimal number. For now we will use integers to make life easier. Also note the sample space of \(X\) does not have an upper bound so plotting X by hand would be really difficult.

We now know that all probability functions (mass or density) start with the letter d in R. The function for exponential density is dexp().

We can use this function to calculate densities as follows:

dexp(x = 0, rate = 0.1)
## [1] 0.1
dexp(x = 1, rate = 0.1)
## [1] 0.09048374
dexp(x = 2, rate = 0.1)
## [1] 0.08187308

Rather than doing these one by one, we can use

dexp(x = 0:10, rate = 0.1)
##  [1] 0.10000000 0.09048374 0.08187308 0.07408182 0.06703200 0.06065307
##  [7] 0.05488116 0.04965853 0.04493290 0.04065697 0.03678794

to calculaute \(f(x)\) for \(X \in \{0, 1, 2, .... 9, 10\}\).Note that this is a vector with 11 elements and the first three elements are exactly like the ones we have calculated above. We can also increase the possibilities for \(x\) by changing the endpoint dexp(x = 0:100, rate = 0.1). `

In order to make a ggplot we would have to have a data frame. So in the next code chunk, we will define a data frame that has all the \(x\)s and \(f(x)\)s.

our_data <- data.frame(our_x = 0:100,
                       our_f_x = dexp(x = 0:100, rate = 0.1))

glimpse(our_data)
## Observations: 101
## Variables: 2
## $ our_x   <int> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
## $ our_f_x <dbl> 0.100000000, 0.090483742, 0.081873075, 0.074081822, 0....

Now that we have a data frame we can use it to plot the pdf.

our_data %>% 
  ggplot(aes(x = our_x,
             y = our_f_x)) +
  geom_line()

And what if we wanted to include \(x\)s with decimal numbers? We can use the seq() to generate a sequence that increases at given increments.

seq(from = 0, to = 2, by = 0.1)
##  [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
## [18] 1.7 1.8 1.9 2.0
our_data <- data.frame(our_x = seq(from = 0, to = 100, by = 0.1),
                       our_f_x = dexp(seq(from = 0, to = 100, by = 0.1),
                                      rate = 0.1))

glimpse(our_data)
## Observations: 1,001
## Variables: 2
## $ our_x   <dbl> 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0,...
## $ our_f_x <dbl> 0.10000000, 0.09900498, 0.09801987, 0.09704455, 0.0960...
our_data %>% 
  ggplot(aes(x = our_x,
             y = our_f_x)) +
  geom_line()

Note that \(x\) can go up to \(\infty\)
Also note that I have used variable names such as our_x to distinguish it from the x argument in the aes() function. If you feel comfortable with programming you may want to use better names for your objects.




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