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11.2
11.2.8
Basic Concepts of Probability Theory
643
Probability Distributions
There are several types of probability distributions (analytical models) for describing various types of discrete and continuous random variables. Some of the common
distributions are given below:
Discrete case
Continuous case
Discrete uniform distribution
Binomial
Geometric
Multinomial
Poisson
Hypergeometric
Negative binomial (or Pascal's)
Uniform distribution
Normal or Gaussian
Gamma
Exponential
Beta
Rayleigh
Weibull
In any physical problem, one chooses a particular type of probability distribution
depending on (1) the nature of the problem, (2) the underlying assumptions associated
with the distribution, (3) the shape of the graph between f (x) or F (x) and x obtained
after plotting the available data, and (4) the convenience and simplicity afforded by the
distribution.
Normal Distribution.
The best known and most widely used probability distribution
is the Gaussian or normal distribution. The normal distribution has a probability density
function given by
f X (x) = _
1
eŠ1 /2[(xŠµ X)/_ X ]2
,
Š_ < x < _
(11.45)
2__X
where µX and _X are the parameters of the distribution, which are also the mean and
standard deviation of X, respectively. The normal distribution is often identified as
N (µ X, _ X).
Standard Normal Distribution.
A normal distribution with parameters µX =
=
1,
called
the
standard
normal
distribution,
is denoted as N(0, 1). Thus the density
•X
function of a standard normal variable (Z) is given by
1
f Z (z) = _
Š(z 2/2)
e
,
Š_ < z < _
0 and
(11.46)
2_
The distribution function of the standard normal variable (Z) is often designated as
_ (z) so that, with reference to Fig. 11.4,
_(z
1)
=p
and
z1 = _
Š1
(p)
(11.47)
where p is the cumulative probability. The distribution function N(0, 1) [i.e., _(z)] is
tabulated widely as standard normal tables. For example, Table 11.1, gives the values
of z, f (z), and _(z) for positive values of z. This is because the density function is
symmetric about the mean value (z = 0) and hence
f (Šz) = f (z)
(11.48)
_(Šz) = 1 Š _(z)
(11.49)
644
Stochastic Programming
Figure 11.4
Standard normal density function.
By the same token, the values of z corresponding to p < 0.5 can be obtained as
Š1
(p) = Š_Š
1(
Š p)
Notice that any normally distributed variable (X) can be reduced to a standard normal
variable by using the transformation
z=_
z=
1
x Š µX
(11.50)
(11.51)
•X
For example, if P (a < X _ b) is required, we have
P (a < X _ b) =
•X
1
_
b
•
2_
2
eŠ (1 / 2)[(xŠµ X)/_ X ] dx
(11.52)
a
dz , Eq. (11.52) can be rewritten as
By using Eq. (11.51) and dx = _X
P (a < X _ b) = _
(bŠµ X)/_X
•
1
2_
(aŠµ X)/_X
2
eŠz
dz2
/
(11.53)
This integral can be recognized to be the area under the standard normal density curve
and hence
between (a Š µ X)/_X and (b Š µ X)/_X
P (a < X _ b) = _
ffi b Š µ
•X
X
,
Š_
ffi a Š µX ,
•X
(11.54)
Example 11.4 The width of a slot on a duralumin forging is normally distributed.
The specification of the slot width is 0.900 ± 0.005. The parameters µ = 0.9 and
• = 0 .003 are known from past experience in production process. What is the percent
of scrap forgings?
SOLUTION
given by
If X denotes the width of the slot on the forging, the usable region is
0 .895 _ x _ 0.905
11.2
Table 11.1
Basic Concepts of Probability Theory
Standard Normal Distribution Table
z
f (z)
_(z)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.5
4.0
4.5
5.0
0.398942
0.396952
0.391043
0.381388
0.368270
0.352065
0.333225
0.312254
0.289692
0.266085
0.241971
0.217852
0.194186
0.171369
0.149727
0.129518
0.110921
0.094049
0.078950
0.065616
0.053991
0.043984
0.035475
0.028327
0.022395
0.017528
0.013583
0.010421
0.007915
0.005952
0.004432
0.000873
0.000134
0.000016
0.0000015
0.500000
0.539828
0.579260
0.617912
0.655422
0.691463
0.725747
0.758036
0.788145
0.815940
0.841345
0.864334
0.884930
0.903199
0.919243
0.933193
0.945201
0.955435
0.964070
0.971284
0.977250
0.982136
0.986097
0.989276
0.991802
0.993790
0.995339
0.996533
0.997445
0.998134
0.998650
0.999767
0.999968
0.999996
0.9999997
and the amount of scrap is given by
scrap = P (x _ 0.895) + P (x
0.905)
In terms of the standardized normal variable,
scrap = P
ffi
Z_
,
ffi
Š0 9 +. 0
+P Z
.895
0
.003
= P (Z _ Š1.667) + P (Z
+1.667)
,
Š0 9 +. 0
.905
0
.003
645
646
Stochastic Programming
= [1 Š P (Z _ 1.667)] + [1 Š P (Z _ 1.667)]
= 2 0.
Š 2P (Z _ 1.667)
= 2 0.
Š 2(0.9525) = 0.095
= 9 5 %.
Joint Normal Density Function.
linear function, Y = a 1X 1 + a
mean
2X
2
Y =a
nXn
and variance
If X 1, X 2, . . . , X n follow normal distribution, any
+ · · · + a nX n, also follows normal distribution with
1X 1
+a
2X 2
+···+a
(11.55)
Var(Y ) = a 12 Var(X1 ) + a2 Var(X 2 ) + · · · + a2 Var(X n
(11.56)
2
n
)
if X 1, X 2, . . . , X n are independent. In general, the joint normal density function
for
n-independent random variables is given by
n
02 1
1
1_ / x kŠ
exp .Š
f X 1,X 2,...,Xn (x 1, x 2, . . . , x n) =
2
Xk
2
( 2_) n_ 1_ 2 · · ·
k=1
_
_n
•k
= fX1 (x
(x 2) · · · (x
fXn
n)
where _ i = _ Xi. If the correlation between the random
variables X k and X
the joint density function is given by
fX 1,X 2,...,Xn
=_
(x 1, x
x n)
1
2,
(11.57)
j
is not zero,
...,
-
n
exp .Š
( 2_)
n|K|
1)fX2
1__
2
1
n
{KŠ 1} jk(x
j
Š X j)(x
k
Š X k)2
(11.58)
j= 1 k=1
where
= E[(x
KX j Xk = Kjk
=
•
•_
Š X j)(x
•
(x
Š_
j
j
k
Š X j)(x
Š X k)]
k
Š X k)f
Š_
= convariance between X
K = correlation matrix = 3
j
X j ,Xk
(x j , x
jdxk
k)dx
and Xk
- K11
K12
3K21
3
.
3 ..
.
K
K22
K
2n
•··
•··
K1n 1
K2n44
4
4
• · · Knn2
1n
Š1
and {KŠ }jk1 = jkth element of K
. It is to be noted that K
X j Xk =
2
• Xj for j = k in case there is no correlation between X j and X k.
(11.59)
0 for j _= k and =