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 and Y:=\max(X_1, X_2,...,X_n). I need to find the cdf and pdf of Y and Z.



Cdf of Y is $$F_Y(x)=P(Y
Then f_Y(x)=F'_Y(x)=na^{n-1}.



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 f_Z(x)=F'_Z(x)=n[1-a]^{n-1}.



Answer



Let n \in \mathbb N and X_1, X_2,...,X_n be \sim^{iid} Unif(0,1).



Define



X_i:=\min(X_1, X_2,...,X_n)



X_a:=\max(X_1, X_2,...,X_n).



What are the distribution and densities of those?







Let c \in \mathbb R.



We have for X_a



$$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



F_{Y}(c) = (a)^n




\to f_{Y}(a) = n(a)^{n-1} a'(c)



\to f_{Y}(a) = n(a)^{n-1} 1_{c \in [0,1]}






We have for X_i



P(X_i < c) = 1 - P(X_i \ge c)




By independence, we have



P(X_i \ge c)=P(X_1 \ge c)P(X_2 \ge c)...P(X_n \ge c)



By identical distribution, we have



=[P(X_1 \ge c)]^n



=[1-P(X_1 < c)]^n := [1-a(c)]^n




Hence we have



F_{X_i}(c) = 1-[1-a(c)]^n



\to f_{X_i}(c) = -n[1-a(c)]^{n-1}(-a'(c))



\to f_{X_i}(c) = n[1-a(c)]^{n-1} 1_{c \in [0,1]}


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