1. Given a random variable X that has a t(ν) distribution, determine the value of x0 such that P (|X| x0) = 0.025 for (i) ν = 5; (ii) (i) ν = 25; (iii) ν = 50; (iv) ν = 100. Compare your results with the single value of x0 such that P (|X| x0) = 0.025 for a standard normal random variable X.

2. Plot on the same graph the pdfs for a Cauchy C(5, 4) random variable and for a Gaussian N (5, 4) random variable. Compute the probability P (X ≥ μ + 1.96σ) for each random variable, where, for the Gaussian random variable, μ and σ are the mean and standard deviation (positive square root of the variance) respectively; for the Cauchy distribution, μ and σ are, the location and scale parameters, respectively.