Question

Consider the random variable Z defined as follows: for z=c for z 1 2K fz(z) for zc otherwise 0 for some constant c> 1 (a) Determine the value of K. Z2 (b) Determine the distribution of the random variable W (c) Find the moment generating function of Z when c 2 (d) Using the moment generating function, determine the mean and the variance of Z when c 2

Answer 1

iren the distributer random vazi able for 2-c 1K = (2) 2 C ca) )is Proper a (2)3 41 1 24 The fmf of 2 is f2(2)= 12 7

Th distYbuhn w is Wニ 2」or い2 V for wzc2 c2 ) w2 W 7I O.w c) Th momerd gemezaing funchon andom Nai able 2 is givenby M2C) = E (e) tz

-tc 2-C, C -t C + when c 2 e + e The momemt Generaun 2 is Rating femchion -2t 2 t M2Ct) diffeentintiny equation with epect to t wr get 2t e MzLH t e

zt t + = (3 at to 2 (1-,3..) 주+두+등- = (9)3w 시~ (o)W diffceenliatin equatim wirt 9et we - 2t e dt 2t 2t

,2t t)e e+et at to, e e e + 2. The rariance is Var (z)m) - 2 2+ 2 2+ 1 4 I+8 4 Var (2) 9 4 ( |