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Task 1: Simulating from a logistic distribution Use the inverse transform method to construct a simulator for the logistic(a, b) distribution with cdf 1 F (x) = where a = 1 and b = 1. Truc incan and Variance a are a and x2b2/3 respectively. The problem Formulate the inverse transform method algorithm to simulate froma logistic distribution. Recall you will need to find the inverse of the cumulative distribution 1000 variates from the logistic distribution using your algorithtn. # [Place code here] Report the following Present the inverse transform method formula for F-1 (x) [Answer here] Present the incan and Variance of the 1000 Variates you simulate Compare these empirical values to the truth. Present a histogram of your 1000 Variates Plot the empirical cumulative distribution function of your 1000 Variates. Overlay the truc cdf on the plot and compare In R, ecdf function will plot the empirical cdf. For the truc cdf, compute the cdf ((x) over the sequence seq(-10, 10, 0.05). Recall the lines function to draw the true cdf on the empirical cdf plot- Make sure to present the truc cdf in a difference color than the empirical coffusing the col option in the lines function. Label the axes on the plot. [Answer here: interpret the summaries and plots comparing the empirical results with the truth] Task 2: Simulating circles Dobrow 6.60:Let R - untform(1, 4). Let A denote the arca of the crecle of radius R. Wc will simulate R and then in turn obtain Variates for the area A. The problem Simulate the area of the circle A in two ways 1. Simulate 100,000 Variates for R and then compute the arca of the circle. 2. Use the inverse transform method to simulate directly from A. # [Place code here] Report the following Present the incan and standard deviation of thearca Present a histogram of the Variates A from the inverse transform method simulation (problem 2 above). Use the sequence seq(pi, 16*pi, by-pi) as the breaks in the histogram. Remember to set freq - FALSE asa histogram option for a density plot- Overlay the truc pdf of Aon the histogram. Use the same sequence seq(pi, 16*pi, by-pi) over which to compute the pdf. The lines function will overlay this true pdf on the histogram plot- [Answer here: interpret the summaries and plots comparing the empirical results with the truth]

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1.

# ## Simulating 1000 variates
x=runif(1000,0,1);
a=1;
b=1;
y=a+b*log((x)/(1-x));

## finding empirical and true mean variance
mean_empirical=mean(y);
var_empirical=sd(y)^2;
true_mean=a;
true_var=(pi^2)*(b^2)/3;

### histogram
h<-hist(y, breaks=12, col="red",ylim=c(0, 300), xlab="x",
main="Histogram of simulated variates")

## plotting ecdf and original cdf
z=seq(-10,10,0.05);...

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