Chapter 15.6, Problem 1P

(a) Find the probability density function $f(x)$ for the position $x$ of a particle which is executing simple harmonic motion on $(-a,a)$ along the $x$ axis. (See Chapter 7, Section 2, for a discussion of simple harmonic motion.) Hint: The value of $x$ at time $t$ is $x=a\cos \omega t.$ Find the velocity $dx/dt;$ then the probability of finding the particle in a given $dx$ is proportional to the time it spends there which is inversely proportional to its speed there. Don’t forget that the total probability of finding the partical somewhere must be 1.

(b) Sketch the probability density function $f(x)$ found in part (a) and also the cumulative distribution function $F(x)$ [see equation (6.4)].

(c)Find the average and the standard deviation of $x$ in part (a).