(b) Considering Î¦2() and introducing polar coordinates in the double integral (a standard trick worth remembering), prove
(c) Show that Ï in (1) is indeed the standard deviation of the normal distribution.
(d) Bernoullis law of large numbers. In an experiment let an event A have probability p(0 < p < 1) and let X be the number of times A happens in n independent trials. Show that for any given > 0,
(e) If X is normal with mean Î¼ and variance Ï2, show that X* = c1X + c2 (c1 > 0) is normal with mean Î¼* = c1Î¼ + c2 and variance Ï*2 = c21Ï2.