Question

Part I: qqplots. This part deals with qaplots of al inds. Let's do some easy experiments, first. Let's take a sample from a normal distribution with mu-2, sigma -3, and then look at the hist and the qqplot of that data: set.seed(3) n 100 # Sample size # Sample from N(mu-2, sigma-3) # Looks normal. But that depends on breaks. x morm(n,2.3) hist(x) qqnorm(x) # This doesn't depend on binsize, and it looks linear. abline(2,3, col-2) What you just got is a qaplot based on a sample, i.e. it's essentially a random "variabe." To get a sense of how much a q run the whole block of code several times but EXcluding the set.seedO line. Remember that all of these qgplots are from a distribution/population that is truly normal, i., hat the population qqplot would look linear. So, now you get a sense of how m variability to allow for when you look at qaplots in the future Also, recall that the intercept and slope of the qaplot are estimates of the mu and sigma of that normal distribution. To intercept and the slope, it's convenient to draw a line on the qqplot. But you want to make sure to focus mostly on the linear portion of the qqplot and mostly ignore the end points. For the set.seed(3) sample, I decided that a line with (intercept, slope) (2,3) is good enough, and so my estimate ofmu and sigma would be 2 and 3, respectively. This being a simulation, we already know that mu-2, eye-ball the sigma 3, and so everything is consistent. Now, let's see what qqplots look like if the population is not normal at all. To that end, a) Write code to take a sample of size 1000 from Exp(lambda1), and make a histogram and a qqplot of the sample. IMPORTANT: use set.seed(3). You will see that the qqplot is not straight at all. In fact, we could have anticipated that it would not be straight by just thinking about three special quantiles: the Oth, the 0.5th and the Ist. b) What are those special quantiles for the standard normal distribution? c) What are those special quantiles for the exponential distribution? Hint: the median of Exp(lambda) is In(2)/lambda.

Please use R

Answer 1

set.seed(3)
n=1000
x=rexp(n,1)
hist(x)
qqnorm(x)
abline(1,1,col=2)

From histogram we can observe that distribution is positively skewed

From below qq-plot we can see line is not straight , hence it is not normal distribution