How to prove P{min(X_1,X_2, ldots ,X_n)=X_i}=(lambda_i)/(lambda_1+cdots+lambda_n) , when X_i is exponentially distributed

steveo963200054 2022-09-14 Answered
How to prove P { min ( X 1 , X 2 , , X n ) = X i } = λ i λ 1 + + λ n , when X i is exponentially distributed
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Answers (2)

Francis Blanchard
Answered 2022-09-15 Author has 12 answers
Assuming the   X j   are independent,
P ( min ( X 1 , X 2 , , X n ) = X i ) = P ( X i X j    for  j i ) = 0 P ( t X j    for  j i | X i = t ) λ i e λ i t d t = 0 j i P ( t X j ) λ i e λ i t d t = 0 λ i e j = 1 n λ j t d t = λ i j = 1 n λ j   .
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cuuhorre76
Answered 2022-09-16 Author has 1 answers
You forgot the assumption X 1 , X 2 , , X n are independent.
So for t>0 and δ t small,
( 0 , )
Thus summing partition of ( 0 , ) into intervals of length δ t, and taking δ t 0,
P ( min ( X 1 , , X n ) = X i ) = 0 λ i e ( λ 1 + + λ n ) t d t = λ i λ 1 + + λ n
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