Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

[ You will need the results about series involving the natural numbers given in
subsection 4.2.5. ]
30.35 The continuous random variablesXandYhave a joint PDF proportional to
xy(x−y)^2 with 0≤x≤1and0≤y≤1. Find the marginal distributions
forXandYand show that they are negatively correlated with correlation
coefficient−^23.
30.36 A discrete random variableXtakes integer valuesn=0, 1 ,... ,Nwith probabil-
itiespn. A second random variableYis defined asY=(X−μ)^2 ,whereμis the
expectation value ofX. Prove that the covariance ofXandYis given by

Cov[X, Y]=

∑N

n=0

n^3 pn− 3 μ

∑N

n=0

n^2 pn+2μ^3.

Now suppose thatXtakes all of its possible values with equal probability, and
hence demonstrate that two random variables can be uncorrelated, even though
one is defined in terms of the other.
30.37 Two continuous random variablesXandYhave a joint probability distribution

f(x, y)=A(x^2 +y^2 ),

whereAis a constant and 0≤x≤a,0≤y≤a. Show thatXandYare negatively
correlated with correlation coefficient− 15 /73. By sketching a rough contour
map off(x, y) and marking off the regions of positive and negative correlation,
convince yourself that this (perhaps counter-intuitive) result is plausible.
30.38 A continuous random variableXis uniformly distributed over the interval [−c, c].
Asampleof2n+ 1 values ofXis selected at random and the random variable
Zis defined as themedianof that sample. Show thatZis distributed over [−c, c]
with probability density function

fn(z)=

(2n+1)!

(n!)^2 (2c)^2 n+1

(c^2 −z^2 )n.

Find the variance ofZ.
30.39 Show that, as the number of trialsnbecomes large butnpi=λi,i=1, 2 ,...,k−1,
remains finite, the multinomial probability distribution (30.146),

Mn(x 1 ,x 2 ,...,xk)=

n!

x 1 !x 2 !···xk!

px 11 px 22 ···pxkk,

can be approximated by a multiple Poisson distribution withk−1 factors:

M′n(x 1 ,x 2 ,...,xk− 1 )=

∏k−^1

i=1

e−λiλxii

xi!

.

(Write

∑k− 1

i pi=δand express all terms involving subscriptkin terms ofnand

δ, either exactly or approximately. You will need to usen!≈n[(n−)!] and

(1−a/n)n≈e−afor largen.)

(a) Verify that the terms ofM′nwhen summed over all values ofx 1 ,x 2 ,...,xk− 1

adduptounity.

(b) Ifk=7andλi=9foralli=1, 2 ,...,6, estimate, using the appropriate

Gaussian approximation, the chance that at least three ofx 1 ,x 2 ,...,x 6 will

be 15 or greater.

30.40 The variablesXi,i=1, 2 ,...,n, are distributed as a multivariate Gaussian, with
meansμiand a covariance matrixV.IftheXiare required to satisfy the linear