a Let Z N 0 1 Write the density fZx and verify that its val

a) Let Z ? N (0, 1). Write the density fZ(x) and verify that its value is positive for all real numbers.

b) What is the Supp [Z ]?

c) Write FZ(1.2345) using an integral (FZ is the CDF of the standard normal).

d) What is P (Z ? [?1, 1])?

e) What is P (Z ? [?2, 2])?

f) WhatisP(Z?[?3,3])?

g) Draw the density fZ (x) and be careful to denote the x-axis like we did in class. Then illustrate the empirical rule just like we did in class. I want to see the 68, 95, 99.7%’s denoted as areas.

h) Draw ?(x) := FZ (x), i.e. the CDF of the standard normal r.v. Use the same scale as your drawing of the PDF of Z.

j) What is P (Z > 1)?

k) What is P (Z ? (?3, 1))?

Solution