Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS 0 0 0. 2 0. 4 0. 6 0. 8 1 1 2 3 4 y g(y) μ=0,σ=0 μ=0,σ=0. 5 μ=0,σ=1. 5 μ=1,σ=1 Figure 30 ...

PROBABILITY 0 0 246810 12 14 16 18 20 0. 2 0. 4 0. 6 0. 8 1 r=1 r=2 r=5 r=10 x f(x) Figure 30.16 The PDFf(x) for the gamma distr ...

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS 30.9.4 The chi-squared distribution In subsection 30.6.2, we showed that ifXis Gaussian ...

PROBABILITY 0 0 0. 2 0. 4 0. 6 0. 8 − 4 − 224 x f(x) x 0 =0, x 0 =0, x 0 =2, Γ=1 Γ=1 Γ=3 Figure 30.17 The PDFf(x) for the Breit– ...

30.10 THE CENTRAL LIMIT THEOREM and its mean and variance are given by E[X]= a+b 2 ,V[X]= (b−a)^2 12 . 30.10 The central limit t ...

PROBABILITY whereMXi(t)istheMGFoffi(x). Now MXi ( t n ) =1+ t n E[Xi]+^12 t^2 n^2 E[Xi^2 ]+··· =1+μi t n +^12 (σ^2 i+μ^2 i) t^2 ...

30.11 JOINT DISTRIBUTIONS consult one of the many specialised texts. However, we do discuss the multinomial and multivariate Gau ...

PROBABILITY 30.11.2 Continuous bivariate distributions In the case where bothXandYare continuous random variables, the PDF of th ...

30.12 PROPERTIES OF JOINT DISTRIBUTIONS 30.11.3 Marginal and conditional distributions Given a bivariate distributionf(x, y), we ...

PROBABILITY
Show that ifXandYare independent random variables thenE[XY]=E[X]E[Y]. LetusconsiderthecasewhereXandYare continuous ...

30.12 PROPERTIES OF JOINT DISTRIBUTIONS One particularly useful consequence of its definition is that the covariance of twoindep ...

PROBABILITY
A biased die gives probabilities^12 p,p,p,p,p, 2 pof throwing1, 2, 3, 4, 5, 6respectively. If the random variableXi ...

30.12 PROPERTIES OF JOINT DISTRIBUTIONS More generally, we find (fora,bandcconstant) V[aX+bY+c]=a^2 V[X]+b^2 V[Y]+2abCov[X, Y]. ...

PROBABILITY
A card is drawn at random from a normal 52 -card pack and its identity noted. The card is replaced, the pack shuffl ...

30.13 GENERATING FUNCTIONS FOR JOINT DISTRIBUTIONS As would be expected,Xis uncorrelated with eitherWorY, colour and face-value ...

PROBABILITY Finally we note that, by analogy with the single-variable case, the characteristic function and the cumulant generat ...

30.15 IMPORTANT JOINT DISTRIBUTIONS where J≡ ∂(x 1 ,x 2 ...,xn) ∂(y 1 ,y 2 ,...,yn) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∂x 1 ∂y 1 ... ∂xn ∂y 1 ...

PROBABILITY 30.15.1 The multinomial distribution The binomial distribution describes the probability of obtainingx‘successes’ fr ...

30.15 IMPORTANT JOINT DISTRIBUTIONS (i) The probability of picking six tickets of the same colour is given by Pr (six of the sam ...

PROBABILITY and thus, using (30.135), we obtain Cov[Xi,Xj]=E[(Xi−μi)(Xj−μj)] = (A−^1 )ij. HenceAis equal to the inverse of the c ...