Final answer:
The density function of the kinetic energy can be found by determining the probability distribution of the random velocity v and substituting it into the formula for kinetic energy.
Step-by-step explanation:
The density function of the kinetic energy can be found by determining the probability distribution of the random velocity v and then substituting it into the formula for kinetic energy. Since v is normally distributed with mean μ = 0 and standard deviation σ, the probability density function (PDF) of v is given by:
f(v) = (1/σsqrt(2π)) * e^((-v^2)/(2σ^2))
To find the density function of the kinetic energy e, we substitute the expression for v into the formula for kinetic energy:
e = (1/2)mv^2 = (1/2)m(μ + σz)^2
where z is a standard normal random variable. By applying the change of variable formula for probability density functions, we can find the density function of e.