Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

x

f(x) F(x)

2 p

p

1

2 p

1

1

2 3 4 5 6 123 4 5 6

(a) (b)

Figure 30.6 (a) A typical probability function for a discrete distribution, that

for the biased die discussed earlier. Since the probabilities must sum to unity

we requirep=2/13. (b) The cumulative probability function for the same

discrete distribution. (Note that a different scale has been used for (b).)

HenceF(x) is a step function that has upward jumps ofpiatx=xi,i=

1 , 2 ,...,n, and is constant between possible values ofX. We may also calculate

the probability thatXlies between two limits,l 1 andl 2 (l 1 <l 2 ); this is given by

Pr(l 1 <X≤l 2 )=

∑

l 1 <xi≤l 2

f(xi)=F(l 2 )−F(l 1 ), (30.41)

i.e. it is the sum of all the probabilities for whichxilies within the relevant interval.

A bag contains seven red balls and three white balls. Three balls are drawn at random

and not replaced. Find the probability function for the number of red balls drawn.

LetXbe the number of red balls drawn. Then

Pr(X=0)=f(0) =

3

10

×

2

9

×

1

8

=

1

120

,

Pr(X=1)=f(1) =

3

10

×

2

9

×

7

8

×3=

7

40

,

Pr(X=2)=f(2) =

3

10

×

7

9

×

6

8

×3=

21

40

,

Pr(X=3)=f(3) =

7

10

×

6

9

×

5

8

=

7

24

.

It should be noted that

∑ 3

i=0f(i) = 1, as expected.

30.4.2 Continuous random variables

A random variableXis said to have acontinuousdistribution ifXis defined for a

continuous range of values between given limits (often−∞to∞). An example of

a continuous random variable is the height of a person drawn from a population,

which can takeanyvalue (within limits!). We can define theprobability density

function(PDF)f(x) of a continuous random variableXsuch that

Pr(x<X≤x+dx)=f(x)dx,