Starting on january 1 we measure the noontime temperature

Question

A simple model (in degrees Celsius) for the daily temperature process C(t) of Example 10.3 is

Where X₁, X₂,... is an iid random sequence of Gaussian (0, 1) random variables.

(a) What is the mean E[C_n]?

(b) Find the autocovariance function C_C[m, k].

(c) Why is this model overly simple?

Example 10.3

Starting on January 1, we measure the noontime temperature (in degrees Celsius) at Newark Airport every day for one year. This experiment generates a sequence, C(1),C(2), . . . ,C(365), of temperature measurements. With respect to the two kinds of averages of stochastic processes, people make frequent reference to both ensemble averages such as "the average noontime temperature for February 19," and time averages, such as the "average noontime temperature for 1923."