Page 234 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 217

hours; its density function and distribution function are given by Equations
(7.52) and (7.55). The desired result is, using Equation (7.55),

; † 9; 6†
P X 1†ˆ F X 1†ˆ ˆ :
† 9†

Now, (9) ˆ 8!, and the incomplete gamma function (9, 6) can be obtained by

table lookup. We obtain:
P X 1†ˆ 0:153:

An alternative computational procedure for determining P(X 1) inExample
7.7 can be found by noting from Equation (7.63) that random variable X can be
represented by a sum of independent random variables. Hence, according to
the central limit theorem, its distribution approaches that of a normal random
variable when is large. Thus, provided that is large, computations such as
that required in Example 7.7 can be carried out by using Table A.3 for normal
random variables. Let us again consider Example 7.7. Approximating X by
a normal random variable, the desired probability is [see Equation (7.25)]

1 m X
P X 1†' PU ;
X
where U is the standardized normal random variable. The mean and standard
deviation of X are, using Equations (7.57),
9
m X ˆ ˆ
6
3
ˆ ;
2
and

1=2 3
X ˆ ˆ
6
1
ˆ :
2
Hence, with the aid of Table A.3,

P X 1†' P U 1†ˆ F U 1†ˆ 1 F U 1†
ˆ 1 0:8413 ˆ 0:159;

which is quite close to the answer obtained in Example 7.7.

TLFeBOOK