da luca.barletta » 17/11/2007, 10:20
(a) Per il CLT si ha:
$((sum_(i=1)^n X_i)/n-1)/(1/sqrtn)rarr Z$ dove $Z~ccN(0,1)$
(b) E' noto che date v.a. $X_i~Poisson(lambda_i)$ indip., allora $sum_(i=1)^n X_i ~ Poisson(sum_i lambda_i)$. Si sfutta questo fatto per notare che:
$lim_(nrarr+infty) e^(-n)(1+n+n^2/2+...+n^n/(n!))=lim_(nrarr+infty) Pr[sum_(i=1)^n X_i <= n]=$
$=lim_(nrarr+infty) Pr[sum_(i=1)^n X_i -n <= 0] = lim_(nrarr+infty) Pr[(sum_(i=1)^n X_i -n)/(sqrtn) <= 0]=$
$=(CLT)=Pr[Z<=0]=1/2$
Frivolous Theorem of Arithmetic:
Almost all natural numbers are very, very, very large.