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# HW3 - If X is exponential with parameter such that fX(x =...

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If X is exponential with parameter λ , such that f X ( x ) = λ exp( - λ x ) , show that the ‘ceiling’ of X (i.e., the next integer, so that the integer of 1.1 is 2, the integer of 2.4 is 3 etc) Y has geometric distribution with parameter p = 1 - exp( - λ ) . Hint: Consider P ( Y = 1) = P (0 < X < 1) , generalize for any i , and compare with the pmf of the geometric which is P ( Y = i ) = p (1 - p ) i - 1 . If X 1 , . . . , X n are exponentially distributed such that f X i ( x ) = λ exp( - λ x ) , show that the distribution of Y = min( X 1 , . . . , X n ) is exponential with pa- rameter n λ , so that f Y ( y ) = n λ exp( - n λ y ) . Hint: In order for min( X 1 , . . . , X n ) > y , we need ALL of them to be GREATER than y . What’s the probability that none are less than
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