When solving problems below, it makes sense to keep in mind (2.4.3).

(a) Let X be standard normal. What is the distribution of −X? Generalize the problem considering a normal r.v. with zero mean. If X is normal with mean m and variance σ², then what is the distribution of −X?

(b) In the same picture, sketch the graphs of the normal distributions with the unit variance and the means equal to 1 and −1, respectively.

(c) In the same picture, sketch the graphs of the normal distributions with zero mean and the standard deviations equal to 1 and 2, respectively. (Advice: Compare the values of the two densities at x = 0 and for large x’s.)