Wednesday, 29 April 2015

probability - Cdf and Pdf of independent random variables(iid)



Let X1,X2,...,Xn be independent random variables, each having a uniform distribution over (0,1). Let Z:=min(X1,X2,...,Xn) and Y:=max(X1,X2,...,Xn). I need to find the cdf and pdf of Y and Z.



Cdf of Y is $$F_Y(x)=P(Y
Then fY(x)=FY(x)=nan1.



Cdf of Z is $$F_Z(x)=P(Zz)= P(X_1>x,X_2>x,...X_n>x)=P(X_1>x)P(X_2>x)\ldots P(X_n>x)=1-a\cdot 1-a \cdot 1-a\cdot\ldots\cdot 1-a=[1-a]^n.$$
Then fZ(x)=FZ(x)=n[1a]n1.



Answer



Let nN and X1,X2,...,Xn be iid Unif(0,1).



Define



Xi:=min(X1,X2,...,Xn)



Xa:=max(X1,X2,...,Xn).



What are the distribution and densities of those?







Let cR.



We have for Xa



$$P(X_a < c) = P(X_1

By independence, we have




$$=P(X_1

By identical distribution, we have



$$=[P(X_1

Hence we have



FY(c)=(a)n




fY(a)=n(a)n1a(c)



fY(a)=n(a)n11c[0,1]






We have for Xi



P(Xi<c)=1P(Xic)




By independence, we have



P(Xic)=P(X1c)P(X2c)...P(Xnc)



By identical distribution, we have



=[P(X1c)]n



=[1P(X1<c)]n:=[1a(c)]n




Hence we have



FXi(c)=1[1a(c)]n



fXi(c)=n[1a(c)]n1(a(c))



fXi(c)=n[1a(c)]n11c[0,1]


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