1. Probability Model Fitting Steps
For a given data x1, x2, …, xN
Plot the histogram with the default bin width
>hist(x, probability=T, …, …)
Also plot the boxplot
Select candidate PDFs based on the histogram and boxplot
Fit the PDFs – i.e., compute the parameters of the PDFs
Evaluate the PDF at the observational points or on a finer grid (for better plotting)
E.g., to fit a Normal PDF to the data
>theta1 = mean(x)
>theta2 = sd(x)
>fitnormpdf = dnorm(sort(x), theta1, theta2)
Similarly fit all the candidate PDFs to the data – using the appropriate commands

2. Goodness of Fit
The goodness of the fitted probability model can be evaluated in two ways – visual and Quantitative
Visual
Overlay the fitted PDF on the histogram
>lines(sort(x), fitnormpdf, col=“red”)
Quantile plots
Compute the quantiles from the fitted PDFs and
plot them agains the empirical quantiles.
If they fall on a straight line then the model is a good fit.
Empirical quantile P_i = (i- a)/(N + 1 – 2a)
i is the ‘rank’ of the observation x_i and a = 0
If the observations are sorted then their ranks are simply the
sequential order
>N = length(x)
>empquant = (1:N)/(N+1)
>fitnormquant = qnorm(empquant, theta1, theta2)
>plot(fitnormquant, sort(x), xlab=“Model Quantiles”, ylab= “Empirical Quantiles”)

3. Goodness of Fit Quantitative
Perform a Kolmogrov-Smirnov (K-S) test on the empirical CDF and the fitted model CDF
>ksnorm = ks.test(x, “pnorm”, mean=theta1, sd=theta2)
This compares the empirical CDF with the fitted Normal PDF
If ksnorm$p.value is ‘greater than or equal’ to 0.05 it implies that
At 95% confidence level the empirical and the fitted distribution are not different
Use the quantitative and visual metrics to decide on the best model

4. Boxplot One step = 1.5*IQR IQR = 33 –26 = 7 °F
One step = 1.5*7 = 10.5 °F
Lower inner fence = 26 – 10.5 = 15.5 °F
Upper inner fence = = 43.5 °F
The whiskers are drawn to the most extreme temperatures inside the inner fences, 37 and 17 °F. The whiskers are therefore shortened to extend only to the last observation within one step beyond either end of the box (“adjacent values”).
One step = 1.5*IQR

5. Histogram
Histogram of Ithaca temperature, January 1987.

6. Histogram and Probability Density Function