Question

Question 6. Explore the Student's t-Distribution Let us use R to become familiar with the Student's t-distribution. For that you may have to use some of the following (R-inbuilt) functions: rt (n, df) ?rt # random generator for distribution # information on rt and its inputs Take 104 samples of a Student's t-distribution with 3 degrees of freedom. Use a morm plot to test the relation to a normal distribution. Comment on your result.

Answer 1

Density 0.0 0.1 0.2 0.3 0

Here the red line is the density plot of random numbers from t distribution and the black one are normal density plot.

Corresponding r code for the above plot

x = rt(10000,3) #Generate 10^4 random numbers from t distribution with df 3.

y = seq(-4, 4, length=10000) # Generate a seq of 10^4 numbers from -4 to 4

hx=dnorm(y) # Calculating the normal density of y

x=sort(x) #sort x in ascending order

plot(y, hx, type="l",ylab="Density",xlab = "") # Plotting normal probability.
lines(x,dt(x,3),col="red") #plotting the density of random number.

Here we see that the red and the black curves are similar. the red one has heavy tails but height is less than the normal density. Hence t distribution is Leptokurtic distribution and symmetric about 0.

As df became large t distribution converge to normal distribution.