(a) Find the points of inflection of the curve of

(a) Find the points of inflection of the curve of (1).

(b) Considering Φ2(ˆž) and introducing polar coordinates in the double integral (a standard trick worth remembering), prove

-и? (2 ди — du = 1. (12) Ф(с) %3D Ф(о) V2п

(c) Show that σ in (1) is indeed the standard deviation of the normal distribution. 

(d) Bernoulli€™s 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 ˆˆ > 0,

+(E-|=-)-1 as n→ 00.

(e) If X is normal with mean μ and variance σ2, show that X* = c1X + c2 (c1 > 0) is normal with mean μ* = c1μ + c2 and variance σ*2 = c21σ2.