This exercise is related to the heat equation (Example 10.15).

(i) Verify that the pdf of N(x, t),

satisfies the heat equation (10.48).
(ii) Verify that the function u(t, x) defined by (10.49) satisfies the heat equation.
(iii) Show, by taking expectations under the integral signs, that (10.50)
implies (10.51); then obtain the heat equation by taking the partial derivatives with respect to t on both sides of (10.51).

Transcribed Image Text:

(y-x) f(y,t,x) = exp -x < y < , /2t 2t