Fundamentals of Probability and Statistics for Engineers

Ex ample 4. 11. Problem: an in sp ection is made of a group of n television
picture tubes. If each passes the inspection with probability p and fails with
probability q (p q 1), calculate the average number of tubes in n tubes that
pass the inspection.
Answer: this problem may be easily solved if we introduce a random variable
Xj to represent the outcome of the jth in sp ection and define

if the jth tube passes inspection;

if the jth tube does not pass in sp ection.

Then random variable Y, defined by

has the desired property that its value is the total number of tubes passing the
inspection. The mean of Xj is

Therefore, as seen from Equation (4.38), the desired average number is given by

We can also calculate the variance of Y. If X 1 ,...,Xn are assumed to be
independent, the variance of Xj is given by

Equation (4.41) then gives

Ex ample 4. 12. Problem: let X 1 ,...,Xn be a set of mutually independent
random variables with a co mmon distribution, each having mean m. Show
that, for every and as n

Note: this is a statement of the law of large numbers. The random variable Y n
can be interpreted as an average of n independently observed random variables
from the same distribution. Equation (4.44) then states that the probability that
this average will differ from the mean by greater than an arbitrarily prescribed

96 Fundamentals of Probability and Statistics for Engineers

‡ ˆ

Xjˆ

1 ;

0 ;



YˆX 1 ‡X 2 ‡‡Xn;

EfXjgˆ 0 …q†‡ 1 …p†ˆp:

mYˆEfX 1 g‡‡EfXngˆnp:

^2 Yˆ^21 ‡‡^2 nˆnpq:

">0, !1,

^2 jˆEf…Xjp†^2 gˆ… 0 p†^2 …q†‡… 1 p†^2 pˆpq:

P

Y

n

m

(^)
(^)
"



! 0 ; whereYˆX 1 ‡‡Xn: … 4 : 44 †

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