Fundamentals of Probability and Statistics for Engineers

Example 4.1.Problem: From Example 3.9 (page64), determine the average
number of cars turning west in a group of n cars.
Answer: we wish to determine the mean of Y for which the mass
function is [from Equation (3.51)]

Equation (4.4) then gives

The sum in this expressions is simply the sum of binomial probabilities and
hence equals one. Therefore,

which has a numerical value since n and q are known constants.

Ex ample 4. 2. Problem: the waiting time X (in minutes) of a customer waiting
to be served at a ticket counter has the density function

Determine the average waiting time.
Answer: referring to Equation (4.5), we have, using integration by parts,

Example 4.3. Problem: from Example 3.10 (pages 65), find the average
resistance of the resistors after screening.

Expectations and Moments 77

,EfYg,

pY…k†ˆ

n

k



qk… 1 q†nk; kˆ 0 ; 1 ; 2 ;...;n:

EfYgˆ

Xn

kˆ 0

kpY…k†ˆ

Xn

kˆ 0

k

n

k

!

qk… 1 q†nk

ˆ

Xn

kˆ 1

n!

…k 1 †!…nk†!

qk… 1 q†nk:

Letk 1 ˆm.Wehave

EfYgˆnq

Xn^1

mˆ 0

n 1

m



qm… 1 q†n^1 m:

EfYgˆnq;

fX…x†ˆ

2e^2 x; forx 0 ;

0 ; elsewhere:

EfXgˆ

Z 1

0

x…2e^2 x†dxˆ

1

2

minute: