Could someone talk me through this question please? I understand some of it, but I'd like some help understanding the rest. Comments are below each line of the answer. Here's the question: Evaluate P(X = k+1)/P(X = k) and hence find the most likely value of X when P(X = k) = r(7.3k/k!) , for all k = 0, 1, 2 ... ; r > 0. Find the value of r and prove that E[X] = 7.3 (where E[X] is the expected value of X). Answer: P(X = k+1)/P(X = k) = r(7.3k+1/(k+1)!) / r(7.3k/k!) = 7.3/(k+1). --> This line I understand just fine. ∴ P(X = k+1) > P(X = k) ⇔ 7.3/(k+1) > 1 ⇔ k < 6.3 --> What I don't understand here is why we can state that P(X = k+1) > P(X = k) ⇔ 7.3/(k+1). Hence increasing for k = 0, 1, 2, ... , 6 and P(7) > P(6) and decreasing for k = 7, 8, 9, ... ∴ Most likely value for k is k = 7. --> I understand the increasing and decreasing parts, but not the P(7) > P(6) or the most likely value of k. ∑∞k=o Pk = 1 => r∑∞k=o (7.3k/k!) = e7.3 = 1 r = e-7.3. --> This line I understand fine. From here, I'm struggling to see what's happening: E[X] = ∑∞k=o ke-7.3(7.3k/k!) = e-7.3(7.3)∑∞k=1 7.3k-1/(k-1)! --> I believe by finding the expected value we are finding the total sum of the weighted means, hence the summation has the form ∑∞k=o k.f(k). However I don't see how the k multiplying the function of k, f(k), has been removed and how the k=1 appears on the summation and k-1 appears in the function, and how the 7.3 finds itself outside the summation as a constant. Let j = k+1, e-7.3(7.3)∑∞j=0 7.3j/(j)! = e-7.3(7.3)e7.3 = 7.3 --> This last line I understand also. Any help you can offer would be greatly appreciated!