(a) Let \(X\) be a random variable with a location-scale t distribution with 3 degrees of freedom, location 100, and scale 10.

i. What is the probability that \(X\) is less than or equal to 110 ? 

ii. What is the 0.25 quantile of \(X\) ?

(b) Generate a sample of 1,000 random numbers following the \((3,100,10)\) location-scale t distribution as above.

Make two q-q plots of the sample side by side, one with a t with 3 degrees of freedom as the reference distribution, and one with a standard normal as the reference distribution. Comment on the plots.

(c) Generate a sample of 1,000 random numbers following the \((6,100,10)\) location-scale t distribution.

Make two q-q plots of the sample side by side, one with a t with 6 degrees of freedom as the reference distribution, and one with a standard normal as the reference distribution. Comment on the plots.