Question

Problem 2. Assume that random variable X has normal distribution with mean 2 and standard deviation of 5 (1) Find the density of random variable Y = X3. (2) Find the mean and variance of random variable Y defined above in (1)

Answer 1

Given x~ N(2,5). 1) We have to find the density of y = x3 7= x3 => x=yo da . y -2/3 dy z žig. . The density function of y is given by gly)= f (y). I do ē 923750 = y* 21.5 I -9-2) 12.25 -2/3 - $25.5.3 – (y-2)/50 -2/3 – ook y <2 2) ECY)= (x3) = Ã (x+p+m) - k{k-p934 (3) *•)?p7+ (3)&=3")p4 po? a of 3.25x2 tot8 = 150+8=156

Now V(Y)= v(x²)= E (x3) ² [E & 37² 866) – EW*)? EGO)= -Par). ES & P40 18-P)*p + ( 9 ) {E="')?p*+ ()&=P)*pu* (5)******** (6) 6-9).p5+ poz = 151(25) + 0 +(.). 3-425740+().254244 0 + 26 - 234,375 + 28, 125+ 6000+ 64 = 268,564