Question

Part I: qqplots This part deals with qqplots of all kinds. Let's do some easy expe riments, 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 x norm(n,2,3) # Sample from N(mu-2, sigma-3) hist(x) # Looks normal. But that depends on breaks. qqnorm(x) # This doesn't depend on binsize, and it looks linear abline(2,3, col-2) how much a qqplot varies, linear. So, now you get a sense of how much istribution. To eye-ball the What you just got is a qqplot based on a sample, i.e., it's essentially a random "variabe. To get a sense of l of these qaplots are from a run the whole block of code several times but EXcluding the set.seed0 line. Remember that a distribution/population that is truly normal, i.e, hat the population qgplot would look 1 variability to allow for when you look at qqplots in the future. Also, recall that the intercept and slope of the qgplot are estimates of the mu and sigma of that normal d t. But you want to make sure to focus mostly on the linear portion intercept and the slope, it's convenient to draw a line on the qqplo of the qqplot and mostly ignore the end points. For the set.seed(3) sample, I decided that a line with (intercept, slope) 2, enough, and so my estimate of mu and sigma would be 2 and 3, respectively. This being a simulation, we already know that mu 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(lambda-1), and make a histogram and a qqplot of the sample. IMPORTANT: use set.seed(3) You will see that the qaplot 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 0th, 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. Just imagine a very, very large sample (i.e., nearly the population). In the qqplot the three numbers in part b would go on the x-axis, and the matching three nunbers in part c would go on the y-axis. And, as you can see, the qqplot would have to look like the one got in part a, i.e., it cannot be linear. These qqplots are called normal qqplots, because the quantiles on the x-axis are those of the standard normal. But the idea of a qgplot can be generalized for testing ANY distribution. For example, if you want to see whether the data come from an exponential distribution, you can put quantiles of *that* distribution on the x-axis. If you get a straight line on that qaplot, then there is evidence that your data have come from an exponential distribution. Although there are R packages that can put on the x-axis quantiles of any distribution you specify, we will do it "by h the way to make the normal qqplot by hand: and." Her is n-100 x-rnorm (n,2,3) # Here is som e data from Normal(2,3), and here is the normal qq plot by hand n-length(x) probs seq(0, 1,length-n) # Probabilities at which quantiles are to be computed Q ; qnorm(probs,0,1 ) # The quantiles of the normal distribution plot(Q, sort(x)) abline(2,3, col-2) # The same line as above # Sample size # qqplot

Solve d,e and f please using R code

Answer 1

i (32-bit) File History Resize Windows 回目回 R Console R R Graphics: Device 2 (ACTIVE) > # (d) sample !from exponentiallaists Untitled -R Editor QQ plot #(d) sample from exponential distribution set.seed (3) lambda--2 ; n=100 x=rexp (100, 2);x #random sample ##qgplot for data qqnorm (x, xlab="empirical quantiles", ylab="theoretical quantiles", main= "QQ plot..) 2 1 01 2 empirical quantiles