Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.2 PROBABILITY O B A 1 A 2 A 3 A 4 Figure 30.5 A collection of traffic islands connected by one-way roads. i=1, 2 , 3 ,4. From ...

PROBABILITY 30.2.3 Bayes’ theorem In the previous section we saw that the probability that both an eventAand a related eventBwil ...

30.3 PERMUTATIONS AND COMBINATIONS We note that (30.27) may be written in a more general form ifSis not simply divided intoAandA ...

PROBABILITY In calculating the number of permutations of the various objects we have so far assumed that the objects are sampled ...

30.3 PERMUTATIONS AND COMBINATIONS the number ofdistinguishablepermutations is only n! n 1 !n 2 !···nm! , (30.32) since theith g ...

PROBABILITY Another useful result that may be derived using the binomial coefficients is the number of ways in whichndistinguish ...

30.3 PERMUTATIONS AND COMBINATIONS may imagine then(distinguishable) objects set out on a table. Each combination ofkobjects can ...

PROBABILITY particles can be distributed among allRquantum states of the system, withniparticles in theith level, is given by W{ ...

30.4 RANDOM VARIABLES AND DISTRIBUTIONS Substituting this expression into (30.37) gives W{ni}=N! ∏R i=1 gi! ni!(gi−ni)! . Such a ...

PROBABILITY x f(x) F(x) 2 p p 1 2 p 1 1 2 3 4 5 6 123 4 5 6 (a) (b) Figure 30.6 (a) A typical probability function for a discret ...

30.4 RANDOM VARIABLES AND DISTRIBUTIONS l (^1) abl 2 x f(x) Figure 30.7 The probability density function for a continuous random ...

PROBABILITY
A random variableXhas a PDFf(x)given byAe−xin the interval 0 <x<∞and zero elsewhere. Find the value of the co ...

30.5 PROPERTIES OF DISTRIBUTIONS In many circumstances, however, random variables do not depend on one another, i.e. they areind ...

PROBABILITY the series is absolutely convergent or that the integral exists, as the case may be. From its definition it is strai ...

30.5 PROPERTIES OF DISTRIBUTIONS Integrating by parts we findA=1/(πa^30 )^1 /^2. Now, using the definition of the mean (30.46), ...

PROBABILITY 30.5.3 Variance and standard deviation Thevarianceof a distribution,V[X], also writtenσ^2 , is defined by V[X]=E [ ( ...

30.5 PROPERTIES OF DISTRIBUTIONS |x−μ|≥c. From (30.48), we find that σ^2 ≥ ∫ |x−μ|≥c (x−μ)^2 f(x)dx≥c^2 ∫ |x−μ|≥c f(x)dx. (30.49 ...

PROBABILITY
A biased die has probabilitiesp/ 2 ,p,p,p,p, 2 pof showing1, 2, 3, 4, 5, 6respectively. Find (i)the mean,(ii)the se ...

30.5 PROPERTIES OF DISTRIBUTIONS We note that the notationμkandνkfor the moments and central moments respectively is not univers ...

PROBABILITY and differentiate it repeatedly with respect toα(see section 5.12). Thus, we obtain dI dα =− ∫∞ −∞ y^2 exp(−αy^2 )dy ...