Question

3. (3 pt) Use R functions to generate 1000 random samples from t-distribution with

15 degrees of freedom. Make a histogram with the samples showing the relative

frequencies. Then overlay a probability density plot over this histogram.

Answer 1

Find below the R code with comments. I assume that you are knowledgeable regarding the theory behind t-dictribution.

x=rnorm(1000) #generate 1000 random values
hist(x, main="1.2a Histogram of 1000 Random N(0,1) Values") #plot histogram

x.grid=seq(-3,3,.01) # create vector of x values

v.sim=(rnorm(1000))^2 + (rnorm(1000))^2 + (rnorm(1000))^2
t.sim = x / sqrt(v.sim/15) # t-distribution of the 1000 random values with 15 degrees of freedom (formulae)

#plotting histogram of t-distribution
hist(t.sim,probability="TRUE", nclass=50,
main="4.2b Histogram of 1000 Random t(df=15) Values\n (density scale)")

#Probability density graph
t.grid=seq(min(t.sim),max(t.sim),.01) #Create vector of x values
density.t.grid=dt(t.grid,df=15) # Compute vector of corresponding density values
lines(t.grid,density.t.grid,col='blue') # use R function lines() to add to current plot

I hope this will give you the desired graph.