Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYThe above results may be extended. For example, if the random variablesX i , i =1, 2,...,n, are distributed as X i ∼ N(µ i ,σi 2 ) then the random variableZ = ∑ i c iX i (where the c i are constants) is distributed as Z ∼ N( ∑ i c iµ i , ∑ i c2 i σ2 i ).30.9.2 The log-normal distributionIf the random variable X follows a Gaussian distribution then the variableY = e X is described by a log-normal distribution. Clearly, if X can take valuesin the range −∞ to ∞, thenY will lie between 0 and ∞. The probability densityfunction for Y is found using the result (30.58). It isg(y) =f(x(y))dx∣ dy ∣ = 1 []σ √ 12π y exp (ln y − µ)2−2σ 2 .We note that µ and σ 2 are not the mean and variance of the log-normaldistribution, but rather the parameters of the corresponding Gaussian distributionfor X. The mean and variance of Y , however, can be found straightforwardlyusing the MGF of X, which reads M X (t) =E[e tX ]=exp(µt + 1 2 σ2 t 2 ). Thus, themean of Y is given byand the variance of Y readsE[Y ]=E[e X ]=M X (1) = exp(µ + 1 2 σ2 ),V [Y ]=E[Y 2 ] − (E[Y ]) 2 = E[e 2X ] − (E[e X ]) 2= M X (2) − [M X (1)] 2 =exp(2µ + σ 2 )[exp(σ 2 ) − 1].In figure 30.15, we plot some examples of the log-normal distribution for variousvalues of the parameters µ and σ 2 .30.9.3 The exponential and gamma distributionsThe exponential distribution with positive parameter λ is given by{λe−λxfor x>0,f(x) =(30.116)0 for x ≤ 0and satisfies ∫ ∞−∞f(x) dx = 1 as required. The exponential distribution occurs naturallyif we consider the distribution of the length of intervals between successiveevents in a Poisson process or, equivalently, the distribution of the interval (i.e.the waiting time) before the first event. If the average number of events per unitinterval is λ then on average there are λx events in interval x, so that from thePoisson distribution the probability that there will be no events in this interval isgiven byPr(no events in interval x) =e −λx .1190