Stochastic Processes in Physics and Chemistry

Chapter03 - Home
Exercises: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 - 15 - 16 - 17 - 18 - 19 - 20 - 21 - 22 - 23 - 24 - 25 - 26 - 27 - 28 - 29 - 30 - 31 - 32 - 33 - 34 - 35 - 36 - 37 - 38 - 39

Exercise. This is the solution to exercise 3.1.1 in the book.

Solution. The n-time average is:

Y (t1)Y (tn) =ψ(t 1)ψ(tn)XnP X(x)dx = Xn i=1nψ(t i)

The variance is:

σ2(t) = (Y (t) Y (t))2 = Y (t)2 Y (t)2

σ2(t) =Y (t)2P X(x)dx Y (t)P X(x)dx2

σ2(t) = ψ(t)2 X2 ψ(t)2 X2 = ψ(t)2σ X2

The autocorrelation function is:

κ(t1,t2) = Y (t1)Y (t2) Y (t1) Y (t2)

κ(t1,t2) = ψ(t1)ψ(t2) X2 ψ(t 1)ψ(t2) X2 = ψ(t 1)ψ(t2)σX2