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 $$F_Y(x)=P(Y
Then fY(x)=F′Y(x)=nan−1.
Cdf of Z is $$F_Z(x)=P(Z
Then fZ(x)=F′Z(x)=n[1−a]n−1.
Answer
Let n∈N 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 c∈R.
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)n−1a′(c)
→fY(a)=n(a)n−11c∈[0,1]
We have for Xi
P(Xi<c)=1−P(Xi≥c)
By independence, we have
P(Xi≥c)=P(X1≥c)P(X2≥c)...P(Xn≥c)
By identical distribution, we have
=[P(X1≥c)]n
=[1−P(X1<c)]n:=[1−a(c)]n
Hence we have
FXi(c)=1−[1−a(c)]n
→fXi(c)=−n[1−a(c)]n−1(−a′(c))
→fXi(c)=n[1−a(c)]n−11c∈[0,1]
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