Question

An investor invests 100 dollars in a stock. The return X (in dollars) after 10 years has
MGF given by M(t) = exp(180t + 800(t^2)).
(a) Find the mean, variance and standard deviation of the return.
(b) Find the probability that the return is more than the initial investment.
(c) Find the probability that the return is in between 150 and 200 dollars.
(d) If Y = e^(X−100), find the density of Y .

Answer 1

het x be the with moment randons variable generating funetan eMt + tora Myct) = then xo NCMO2) Here MxCt) - € 180€ + t2800 = M = 180° 0²=8000 = 1600 @ Mean' p = 180o buch isla Ivan ince o? - 1600 Standard deviahon r= 40 © PC X>100) PC Ya M > 100-180) Zromo, p(2>-2) = 0.9772 using Normal table

(d) y = (C) PC 150 2 XL 200) > PC 100- 180 ZZ C 20 - 180) -40 ap[-0.75 L2L 0.5] so = 0.4648 using table ex-100 et e 100 if xx N(M. o") the ex a Log Noosallmos Bet a logNoonsal ( 180, 1000) if you LN quio's thes axo LN(Mutlna Here a= 100 100 ex o LN( 180 + ene 1600) - LN ( 80 , 1600) 7 ye ex-100 is Logroomal (9 500) cry -100 +93 - 4 40.52T e 2 x 1606