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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 =