Question

R2. The goal in this problem is to create something called a P-P plot (you'll need the result from R1 at a key place). Here's the idea: We start by choosing (say) 100 values for q evenly spaced between 0 and 10 (qvals). We also generate (say) 100000 random numbers from what R claims is an exponential distribution with (say) λ-1/2. Using a for loop, for each value of q, we calculate what percentage of numbers from the exponential distribution we expect to be less than q (see R1), and also, what percentage of numbers from our list of 100000 that actually are less than q. After doing all this, we plot the 100 expected percentages against the 100 actual percentages Think about what graph we should get before you complete the code and then run the below code. Include a rough sketch of your plot and explain why it is shaped the way it is. 1 set.seed (3.1415) # bring our random nesses into alignment # how many values for q 3 ngs 4 lambda = 1/2 6 expectedpercentages - 7 actualpercentages 8 nnums - 9 nums # use seq function to help you # set up vector for expected values # set up vector for actual values # how many random numbers to create # make the random nums 10 11 for i in 1:nqs) 12 expectedpercentages[i]- 13 actualpercentagesi] 14 15 16 plot

Answer 1

If the actual percentages match the theoretical percentages , the points in a PP Plot plot would fall on a $45 \degree$ line.

Here we have drawn a sample from what R claims to be an exponential distribution, and comparing it against actual exponential distribution. Hence the actual percentages should closely match with that of the theoretical/expected percentages and the points would be around $45 \degree$ line.

If x has exponential distribution with parameter $\lambda=\frac{1}{2}$ , the cdf of X is

$P(X<x)=F(x)=1-e^{\lambda x}=1-e^{-\frac{x}{2}}$

The theoretical percentage of numbers from exponential distribution we expect to be less than q is

$\text{Expected Percentages}=P(X<q)\times 100=(1-e^{-\frac{q}{2}})\times 100$

Following is the R code (all statements starting with # are comments)

output is

We can say that the numbers generated from what R claims to be exponential, are indeed exponential as the points lie on a $45 \degree$ line