split univariate distributions into two chapters

_bookdown.yml

@@ -7,7 +7,8 @@ output_dir: published
 rmd_files: [
   "index.Rmd",
   "setup.Rmd",
-  "univariate-distributions.Rmd",
+  "discrete-distributions.Rmd",
+  "continuous-distributions.Rmd",
   "joint-distributions.Rmd",
   "clt.Rmd",
   "data-analysis.Rmd",

continuous-distributions.Rmd

@@ -0,0 +1,205 @@
+# Continous Probability Distributions
+
+## Learning Objectives {-#objectives-continous-distributions}
+
+1. Define and explain the key characteristics of the continous distributions:
+  - normal, 
+  - lognormal, 
+  - exponential, 
+  - gamma, 
+  - chi-square, 
+  - $t$, 
+  - $F$, 
+  - beta and 
+  - uniform on an interval.
+2. Evaluate probabilities and quantiles associated with such distributions.
+3. Generate discrete and continous random variables using the inverse transform method.
+
+## Theory {-#theory-continous-distributions}
+
+**A reminder:** `R` was designed to be used for statistical computing - so it handles **randomness** well! Using `R` we can guarantee reproducibility (and enhance sharability) by using the function `set.seed(seed)`{.R} where `seed` is a single value integer. Using this approach we guarantee the generation of the same sequence of random numbers everytime we call this function. Use `?set.seed`{.R} to learn more about this function.
+
+## In-built probability distributions
+
+**A recap:** `R` has in-built functions for probability distributions:
+
+- **d***\<distribution-name\>* $:=$ **d**ensity ("_PDF_"), *i.e.* $f_X(x)$
+- **p***\<distribution-name\>* $:=$ **p**robability distribution cumulative function ("_CDF_"), *i.e.* $F_X(x) =\boldsymbol{P}(X \leq x)$
+- **q***\<distribution-name\>* $:=$ **q**uantile function, *i.e.* return $x$ such that $\boldsymbol{P}(X \leq x) = p$
+- **r***\<distribution-name\>* $:=$ **r**andom deviates, *i.e.* (*psuedo*) random number generator for a given distribution
+- *Where \<distribution-name\>* $=$ Normal, uniform, lognormal, Student's $t$, Poisson, binormal, Weibull ... see `?distributions()`{.R} for more information
+
+To give some quick examples (we will further explore these in more detail later in this chapter):
+
+| R Code | Definition |
+|-|-|
+| `rnorm(1)`{.R} | Generates $x_1$ where $X \sim \mathcal{N}(0,\,1)$ |
+| `rnorm(y, mean=10, sd=2)`{.R} | Generates $\{y_1,\,y_2,\,\dots\}$ with $Y \sim \mathcal{N}(10,\,2^2)$ |
+| `runif(3, min=5, max=10)`{.R} | Generates $\{z_1,\,z_2,\,z_3\}$  where $Z \sim \mathcal{U}(5,\,10)$ |
+| `dbinom(4, size=5, prob=0.5)`{.R} | Computes $\boldsymbol{P}(X = 4)$ where $X \sim Bin(5,\,0.5)$ |
+| `pgamma(0.2, shape=2, rate=2)`{.R} | Computes $F_Y(0.2)$ where $Y \sim \mathcal{\Gamma}(2,\,2)$, i.e.  $\boldsymbol{P}(Y\leq 0.2)$|
+| `qexp(0.5, rate = 2)`{.R} | Determines smallest value of $z$ for $\boldsymbol{P}(Z \leq z) = 0.5$ where $Z \sim Exp(2)$ |
+
+### Continous probability distributions covered {-}
+
+We will consider how to interact with the following continous probability distributions in `R`:
+
+- Normal
+- Lognormal
+- Exponential
+- Gamma
+- $\chi^2$
+- Student's $t$
+- $F$
+- Beta
+- Uniform (on an interval)
+
+For each distribution above we will determine how to calculate:
+
+- A random deviate following the discrete distribution $X$,
+- The probability density function ("_PDF_"), $P(k_1 \leq X \leq k_2)$ for distribution $X$ over the range $[k_1,\,k_2]$,
+- The cumulative distribution function ("_CDF_"), $P(X \leq k)$, and
+- The quantile function to find $k$ representing the value such that $P(X \leq k) = p$, i.e. the _pth_ percentile.
+
+We will finish off with a plot of the distribution.
+
+## Normal distribution
+
+We start with generating random deviates from the Normal distribution.
+
+If we are interested in the standard normal, `R` helpfully has default argument values for $\mu = 0$ and $\sigma = 0$ so the function call is very concise:
+
+```{r continuous-distributions-normal-1}
+# Generate random deviates
+set.seed(42)
+rnorm(5)
+```
+
+We can also specify our own values of $\mu$ and $\sigma$. With `R` we need to remember that the $\sigma$ argument corresponds to the standard deviation, **not** the variance.
+
+```{r continuous-distributions-normal-2}
+# Generate random deviates
+set.seed(42)
+rnorm(5, 10, 2)
+```
+
+We next look at the cumulative distribution function for a Normal distribution. In `R` we can calculate this using `pnorm(q, mean, sd)`{.R} where:
+
+- _q_ is the quantile of interest,
+- _mean_ is the mean, and
+- _sd_ is standard deviation.
+
+```{r continuous-distributions-normal-3}
+# Calculate P(X <= 2) for X~N(0,1)
+```
+
+Next we want to find the x<sup>th</sup> percentile of $X \sim N(\mu, \sigma)$. We use the quantile function, `qnorm(mu, sigma)`{.R}.
+
+```{r continuous-distributions-normal-4}
+# Find the 99th percentile for X~N(10,2)
+percentile_99 <- qnorm(0.99, 10, 2)
+
+paste0(
+  "The 99th percentile of X~N(10, 2) is ",
+  format(percentile_99, digits = 4),
+  "."
+)
+```
+
+As is customary we finish with a plot of the normal distribution.
+
+```{r continuous-distributions-normal-5}
+# Plot the distribution function of X~Normal(mu =, sigma = )
+```
+
+## Lognormal distribution
+
+```{r continuous-distributions-lognormal-1}
+
+# Plot the distribution function of Y~Lognormal()
+```
+
+## Exponential distribution
+
+```{r continuous-distributions-exponential-1}
+
+# Plot the distribution function of Z~Exp()
+```
+
+## Gamma distribution
+
+```{r continuous-distributions-gamma-1}
+
+# Plot the distribution of X~Gamma()
+```
+
+## $\chi^2$ distribution
+
+```{r continuous-distributions-chi-squared-1}
+
+# Plot the distribution of Y~Chi-Square()
+```
+
+## Student's $t$ distribution
+
+```{r continuous-distributions-t-1}
+
+# Plot the distribution of Z~t()
+```
+
+## $F$ distribution
+
+```{r continuous-distributions-F-1}
+
+# Plot the distribution of X~F()
+```
+
+## Beta distribution
+
+```{r continuous-distributions-Beta-1}
+
+# Plot the distribution of Y~Beta()
+```
+
+## Uniform distribution {#continous-uniform}
+
+```{r continuous-distributions-uniform-1}
+
+# Plot the distribution of Z~Uniform()
+```
+
+## Inverse transform method
+
+The inverse transform method is a way to generate psuedo-random numbers from any probability distribution. 
+
+One possible algorithm is as follows:
+
+1. Generate a random number $u$ from $U \sim \mathcal{U}(0, 1)
+2. Find the inverse of the desired cumulative distribution function, $F^{-1}_X(x)$
+3. Compute $X = F^{-1}_X(u)$
+
+Suppose we wanted to draw 10,000 random numbers from $X \sim Exp(\lambda = 2)$. In order to use the inverse transform method we first need to find the inverse of the CDF. For any $X \sim Exp(\lambda)$, the inverse of the CDF is $\frac{-log(1-x)}{\lambda}$. We can thus use the inverse transform algorithm to generate random deviates following $X \sim Exp(2)$:
+
+```{r continuous-distributions-inverse-transform-1}
+# Step 0 - to guarantee reproducibility
+set.seed(42)
+
+# Step 1 - generate 10,000 random deviates from U[0,1]
+u <- runif(10000)
+
+# Step 2 - find the inverse of the CDF: 1 - exp(-lambda.x)
+# Inverse of CDF = -log(1 - x) / lambda
+
+# Step 3 - compute X using the inverse of the CDF [from step 2] and the random deviates u [from step 1]
+x <- -log(1 - u) / 2
+
+# Plot the resulting x deviates
+library(ggplot2)
+df <- data.frame(x = x) 
+ggplot(df, aes(x=x)) +
+  geom_histogram(binwidth = 0.5, colour="black", fill="white")
+```
+
+## `R` Practice {-#practice-continous-distributions}
+
+We finish with a comprehensive example of an univariate continous distribution question in `R`.