Question

The distribution of the random variable \(X\) in Example \(1.7 .3\) is a member of the log- \(F\) familily. Another member has the cdf $$ F(x)=\left[1+\frac{2}{3} e^{-x}\right]^{-5 / 2}, \quad-\infty

Short answer

The pdf of the given cdf is obtained by differentiating the cdf. Write a function in R to compute this cdf and visually estimate quartiles and median from a plot. Compute the inverse of the cdf and use it to calculate percentiles, which should match the estimates from the plot.

Step-by-step solution

Compute the pdf from the cdf

Compute the derivative of the given cdf function. The derivative of a cdf is the pdf. In this case, use chain rule for differentiation considering \( F(x)=[1+\frac{2}{3} e^{-x}]^{-\frac{5}{2}} \).

Write an R function for the cdf

You will write a function in R that returns the cumulative distribution for the given function. Use the function 'plot' to graph this function over a reasonable interval, and then visually estimate the quartiles and median from the function plot. The function should look like this: \begin{verbatim}cdf_func <- function(x) { return((1+2/3*exp(-x))^(-5/2))}\ #then plot plot(seq(-10,10,by=0.1),sapply(seq(-10,10,by=0.1),cdf_func),type='l')\end{verbatim} The quartiles and median can be estimated visually from this plot.

Compute the inverse of the cdf

Mathematically derive the inverse of the given cdf. The function 'quantile' in R will then be used on this inverse cdf to calculate the percentiles.

Compare the percentiles

Compare the percentiles obtained from the plot inspection in Step 2 with those calculated using the inverse of the function in Step 3. The percentiles should be approximately the same.