Question

Suppose X ∼ N(0, 1).

(1) Explain the density of X in terms of the diffusion process.

(2) Calculate E(X), E(X^2 ), and Var(X).

(3) Let Y = µ + σX. Calculate E(Y ) and Var(Y ). Find the density of Y.

Answer 1

f(x) = | X f(x)da E(X)= уж

= 0

(21) (2π)-121-exp(-2x = (2n)-1/2[-1 + 0] + (2n)-12[0 + 1 ] -(2π)-1/2 = (2π)-12

(21) (2π)-121-exp(-2x = (2n)-1/2[-1 + 0] + (2n)-12[0 + 1 ] -(2π)-1/2 = (2π)-12

(2n)-1-r x2 exp(-1x2) (2r)-12 +exp- -I, exp(Ax*)d, (integrating by parts) = (2π)-12 (0-0) + (0-0)+「 exp(-1x2 (2Z)-1/21 exp(-1x = | (the integral of a pdf over its support is equal to 1) fr(x)dx = 1 E 02-0 var[x]-E[H]-E 2-1-0-1
$Y=\mu+\sigma X\\E(Y)=\mu+\sigma E(X)\\=\mu+\sigma \mu$

Var(Y) o Var(X)

density of Y

$f(y)=\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(X-\mu)}{2\sigma}}$