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(Z
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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