Question

2. Suppose ten particles, each of mass 2 grams, are moving independently, each with a velocity (cm/sec) which is normally distributed with mean 0 and variance 9. Find the distribution of the total kinetic energy of all of these particles. (the kinetic energy of a particle of a mass m 13 point] grams travelling at a velocity v cm/sec is given by mu2 ergs.)

Answer 1

m = 2

v =N(0 , 3^2)

KE = 1/2 m v^2

P(K < k)

= P(1/2 mv^2 < k)

= P(1/2 * 2 v^2 < k)

= P(v^2 < k)

= P( - sqrt(k) < v < sqrt(k))

= $\phi$ (sqrt(k) ) - $\phi$ (-sqrt(k) )

pdf

= d/dk P(K < k)

= d/dk ($\phi$ (sqrt(k) ) - $\phi$ (-sqrt(k) )

= 2 *1/ (2 sqrt(k)) * f(sqrt(k))                           { d/dk $\phi$ (k) = f(k)    , pdf of normal distribution}

= 1/sqrt(k)   f( sqrt(k))

= $\frac{1}{\sqrt{k}} \frac{1}{\sqrt{ 2* \pi * 9}} e^{\frac{-k}{2*9}}$                                                       0 < k < infinity

-e 18 187ん