Advanced High-School Mathematics

SECTION 6.2 Continuous Random Variables 357

6

x

y

χ^2 distribution with four degrees of freedom

χ^2 distribution with one degree of freedom

9. (The Maxwell-Boltzmann density function) The Maxwell-
   Boltzmann distribution comes from a random variable of the form

Y =

√

X 12 +X 22 +X 32 ,

whereX 1 , X 2 , X 3 are independent normal random variables with

mean 0 and variancea^2. Given that the density of theχ^2 -random

variable with three degrees of freedom, show that the density ofY

is given by

fY(t) =

Ã

2

π

Ñ

t^2 e−t

(^2) /(2a (^2) )
a^3
é
.
This distribution is that of the speeds of individual molecules in
ideal gases.^16

10. Using integration by parts, show thatE(χ^2 ) = 1 and that Var(χ^2 ) =
    2 whereχ^2 has one degree of freedom. Conclude that the expected
    value of theχ^2 random variable with n degrees of freedom is n
    and the variance is 2n. We’ll have much more to say about theχ^2
    distribution and its application in Section 6.6.


11. Here’s a lovely exercise.^17 Circles of radius 1 are constructed in the
    plane so that one has center (2rand,0) and the other has center
    (2rand,1). Compute the probability that these randomly-drawn

(^16) It turns out that the the standard distributionais given bya=kT
(in degrees Kelvin),mis the molecular mass, andkis the Boltzmann constantm, whereTis the temperature
k= 1. 3806603 × 10 −^23 m^2 ·kg/s^2 ·K.
(^17) This is essentially problem #21 on the 2008 AMC (American Mathematics Competitions) contest
12 B.