Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

30.11.2 Continuous bivariate distributions

In the case where bothXandYare continuous random variables, the PDF of

the joint distribution is defined by

f(x, y)dx dy=Pr(x<X≤x+dx, y < Y≤y+dy),

(30.128)

sof(x, y)dx dyis the probability thatxlies in the range [x, x+dx]andylies in

the range [y, y+dy]. It is clear that the two-dimensional functionf(x, y) must be

everywhere non-negative and that normalisation requires
∫∞

−∞

∫∞

−∞

f(x, y)dx dy=1.

It follows further that

Pr(a 1 <X≤a 2 ,b 1 <Y≤b 2 )=

∫b 2

b 1

∫a 2

a 1

f(x, y)dx dy.

(30.129)

We can also define the cumulative probability function by

F(x, y)=Pr(X≤x, Y≤y)=

∫x

−∞

∫y

−∞

f(u, v)du dv,

from which we see that (as for the discrete case),

Pr(a 1 <X≤a 2 ,b 1 <Y≤b 2 )=F(a 2 ,b 2 )−F(a 1 ,b 2 )−F(a 2 ,b 1 )+F(a 1 ,b 1 ).

Finally we note that the definition of independence (30.127) for discrete bivariate

distributions also applies to continuous bivariate distributions.

A flat table is ruled with parallel straight lines a distanceDapart, and a thin needle of

lengthl<Dis tossed onto the table at random. What is the probability that the needle

will cross a line?

Letθbe the angle that the needlemakes with the lines, and letxbe the distance from
the centre of the needle to the nearest line. Since the needle is tossed ‘at random’ onto
the table, the angleθis uniformly distributed in the interval [0,π], and the distancex
is uniformly distributed in the interval [0,D/2]. Assuming thatθandxare independent,
their joint distribution is just the product of their individual distributions, and is given by

f(θ, x)=

1

π

1

D/ 2

=

2

πD

.

The needle will cross a line if the distancexof its centre from that line is less than^12 lsinθ.
Thus the required probability is

2

πD

∫π

0

∫^1

2 lsinθ

0

dx dθ=

2

πD

l

2

∫π

0

sinθdθ=

2 l

πD

.

This gives an experimental (but cumbersome) method of determiningπ.