TEAM PROJECT. Normal Distribution. (a) Derive the formula~

Chapter 24, Problem 24.8

(choose chapter or problem)

TEAM PROJECT. Normal Distribution. (a) Derive the formula~ in (6) and (7) from the appropriate normal table. (b) Show that cI>(-:) = I - cI>(:). Give an example. (e) Find the points of inflection of the curve of (1). (d) Considering cI>2(ao) and introducing polarcoordinates in the double integral (a standard trickworth remembering), prove(12) 1 IX 2 cI>(x) = ~ e-ll 12 dll = 1.\. 2'11" -x(e) Show that u in (1) is indeed the standard deviationof the normal distribution. [Use (12).](I) Bernoulli's law oflarge nwnbers.ln an experimentlet an event A have probability p (0 < P < I), and letX be the number of time~ A happens in /I independenttrials. Show that for any given E > 0,as /1-+ x.(g) Transformation. If X is normal with mean J.Land variance u 2 , show that X* = clX + C2 (cI > 0)is normal with mean J.L* = CIJ.L + C2 and varianceU*2 = C1 2U 2 .