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

(b) Considering Î¦^{2}(ˆž) and introducing polar coordinates in the double integral (a standard trick worth remembering), prove

(c) Show that Ïƒ 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) If X is normal with mean Î¼ and variance Ïƒ^{2}, show that X* = c_{1}X + c_{2} (c_{1} > 0) is normal with mean Î¼* = c_{1}Î¼ + c_{2} and variance Ïƒ*^{2} = c^{2}_{1}Ïƒ^{2.}

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