Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.6 FUNCTIONS OF RANDOM VARIABLES functiong(y) for the new random variableY? We now discuss how to obtain this function. 30.6.1 ...

PROBABILITY lighthouse L beam O θ coastline y Figure 30.8 The illumination of a coastline by the beam from a lighthouse.
A ligh ...

30.6 FUNCTIONS OF RANDOM VARIABLES y y+dy dx 1 dx 2 X Y Figure 30.9 Illustration of a functionY(X) whose inverseX(Y) is a double ...

PROBABILITY IfXandYare both discrete RVs then p(z)= ∑ i,j f(xi,yj), (30.60) where the sum extends over all values ofiandjfor whi ...

30.6 FUNCTIONS OF RANDOM VARIABLES where the range of integration is over all possible values of the variablesxi.This integral i ...

PROBABILITY the variablesXandY. For example, ifXandYare continuous RVs then the expectation value ofZis given by E[Z]= ∫ zp(z)dz ...

30.7 GENERATING FUNCTIONS variance of both sides of (30.66), and using (30.68), we find V[Z(X, Y)]≈ ( ∂Z ∂X ) 2 V[X]+ ( ∂Z ∂Y ) ...

PROBABILITY If, as previously, the probability that the random variableXtakes the valuexn isf(xn), then ∑ n f(xn)=1. In the pres ...

30.7 GENERATING FUNCTIONS and differentiating once more we obtain d^2 ΦX(t) dt^2 = ∑∞ n=0 n(n−1)fntn−^2 ⇒ Φ′′X(1) = ∑∞ n=0 n(n−1 ...

PROBABILITY n r=n r Figure 30.10 The pairs of values ofnandrused in the evaluation of ΦX+Y(t). Sums of random variables We now t ...

30.7 GENERATING FUNCTIONS i.e. the PGF of the sum of two independent random variables is equal to the product of their individua ...

PROBABILITY an expression for the PGF ΞS(t)ofSN: ΞS(t)= ∑∞ k=0 ξktk= ∑∞ k=0 tk ∑∞ n=0 hn×coefficient oftkin [ΦX(t)]n = ∑∞ n=0 hn ...

30.7 GENERATING FUNCTIONS The MGF will exist for all values oftprovided thatXis bounded and always exists at the pointt=0whereM( ...

PROBABILITY Scaling and shifting IfY=aX+b,whereaandbare arbitrary constants, then MY(t)=E [ etY ] =E [ et(aX+b) ] =ebtE [ eatX ] ...

30.7 GENERATING FUNCTIONS probability that value ofSNlies in the intervalstos+dsis given by§ Pr(s<SN≤s+ds)= ∑∞ n=0 Pr(N=n)Pr( ...

PROBABILITY so thatCX(t)=MX(it), whereMX(t)istheMGFofX. Clearly, the characteristic function and the MGF are very closely relate ...

30.7 GENERATING FUNCTIONS Comparing this expression with (30.92), we find thatκ 1 =μ,κ 2 =σ^2 and all other cumulants are equal ...

PROBABILITY Distribution Probability lawf(x)MGFE[X] V[X] binomial nCxpxqn−x (pet+q)n np npq negative binomial r+x−^1 Cxprqx ( p ...

30.8 IMPORTANT DISCRETE DISTRIBUTIONS 1 1 1 1 2 2 2 2 3 3 3 3 44 4 4 5 5 5 (^5678910678910) 0 0 0 00 0 0 0 0. 1 0.^1 0. 1 0. 1 0 ...

PROBABILITY For evaluating binomial probabilities a useful result is the binomial recurrence formula Pr(X=x+1)= p q ( n−x x+1 ) ...