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( - λx ) , 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( - λx ) , show that the distribution of Y = min( X 1 , . . . , X n ) is exponential with pa- rameter , so that f Y ( y )= exp( - nλy ) . 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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