Estimation of an exponential parameter

#1
Hi,

I´m having a hard time trying to figure out this.

Exercise
Let X1,..,Xn be i.i.d. Exp(λ) random variables, where λ is unknown.
What is the distribution of min(Xi)? Enter the pdf f_min(x) of min(Xi) in terms of x .

My approach
Since the pdf of an exponential is λ *e^(−λx) and the first term X1=0 regarding the pdf of the distribution.
the pdf would be λ *e^(−λ(0))=λ *1=λ 
 
#2
this is my hint: You can proceed with the definition of Survival Function as below


Immagine.png

Now you can easily get the CDF(z) and the corresponding pdf(z) by simply deriving the CDF with respect to z
Generally speaking, the question has a common answer: the CDF of the min of n iid random variables is given by the following formula

F(min)=1-(1-F)^n
 
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