Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.8 IMPORTANT DISCRETE DISTRIBUTIONS Thus Mi(t)=E [ etXi ] =e^0 t×Pr(Xi=0)+e^1 t×Pr(Xi=1) =1×q+et×p =pet+q. From (30.89), it fo ...

PROBABILITY i=1, 2 ,...,N, is distributed asXi∼Bin(ni,p)thenZ=X 1 +X 2 +···+XNis distributed asZ∼Bin(n 1 +n 2 +···+nN,p), as wou ...

30.8 IMPORTANT DISCRETE DISTRIBUTIONS 2, isr+x−^1 Cx. Therefore, the total probability of obtainingxfailures before the rth succ ...

PROBABILITY where in the last linep=M/Nandq=1−p. This is called thehypergeometric distribution. By performing the relevant summa ...

30.8 IMPORTANT DISCRETE DISTRIBUTIONS discrete random variablesXdescribed by a Poisson distribution are the number of telephone ...

PROBABILITY t+∆tis given by Px(t+∆t)=Px(t)(1−λ∆t)+Px− 1 (t)λ∆t. Rearranging the equation, dividing through by ∆tand letting ∆t→0 ...

30.8 IMPORTANT DISCRETE DISTRIBUTIONS 1 1 12 2 2 3 3 3 4 4 4 5 5 5 6 6 7 7891011 0 0 0 0 0 0 0. 1 0. 1 0. 1 0. 2 0. 2 0. 2 0. 3 ...

PROBABILITY from which we obtain M′X(t)=λeteλ(e t−1) , M′′X(t)=(λ^2 e^2 t+λet)eλ(e t−1) . Thus, the mean and variance of the Poi ...

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS expression for the PDF of thisZ does exist, but it is a rather complicated combination o ...

PROBABILITY Distribution Probability lawf(x)MGF E[X] V[X] Gaussian 1 σ √ 2 π exp [ − (x−μ)^2 2 σ^2 ] exp(μt+^12 σ^2 t^2 ) μσ^2 e ...

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS − 6 − 4 − 2 234 6 8 10 12 0. 1 0. 2 0. 3 0. 4 σ=1 σ=2 σ=3 μ=3 Figure 30.13 The Gaussian ...

PROBABILITY It is usual only to tabulate Φ(z)forz>0, since it can be seen easily, from figure 30.14 and the symmetry of the G ...

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS Φ(z) .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 .5000 .5040 .5080 .5120 .5160 .5199 .52 ...

PROBABILITY
SawmillAproduces boards whose lengths are Gaussian distributed with mean209.4 cm and standard deviation5.0 cm. A bo ...

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS Now using Φ(−z)=1−Φ(z)gives Φ ( μ− 140 σ ) =1− 0 .030 = 0. 970. Using table 30.3 again, ...

PROBABILITY xf(x) (binomial) f(x) (Gaussian) 0 0.0001 0.0001 1 0.0016 0.0014 2 0.0106 0.0092 3 0.0425 0.0395 4 0.1115 0.1119 5 0 ...

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS the Gaussian distribution. Explicitly, Pr(X=x)≈ 1 σ √ 2 π ∫x+0. 5 x− 0. 5 exp [ − 1 2 (u ...

PROBABILITY Stirling’s approximation for largexgives x!≈ √ 2 πx (x e )x implying that lnx!≈ln √ 2 πx+xlnx−x, which, on substitut ...

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS In fact, almost all probability distributions tend towards a Gaussian when the numbers i ...

PROBABILITY The above results may be extended. For example, if the random variables Xi,i=1, 2 ,...,n, are distributed asXi∼N(μi, ...