Probability question dealing with expected value shown below

a random variable X has a distribution ae^(-ax),  Let [x] denote the greatest integer function applied to the real number ?, that is, the largest integer among those not exceeding ?.
What is the correct expression for the expected value of ? =[x]? the answer should involve a

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  • how did you get (e^(a) -1)e^(-a(w+1)) actually?

  • It's the derivative of 1−e ^(−a(w+1)). Chain rule.

  • if you simplify 1-e^(-aw-a) to 1-(e^-aw)(e^-a) then the derivative with respect to w is (ae^(-aw))(e^-a) right? says the same thing on symbolab https://www.symbolab.com/solver/step-by-step/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20x%7D%5Cleft(1-e%5E%7B-ax-a%7D%5Cright)?or=input

  • You’re right. I realized you can just find the PMF directly with an integral. I’ve edited my answer.

  • okay, thanks

The answer is accepted.