So I was bored; and decided to do some math about pw. It happens to be a beautiful result.Recently, some people are asking to lower the proc. rate of spirit blackhole on forum; and the topic interests me.So I was thinking about a random problem, which is not really related to the topic to most of the people here. Hence, this thread was created in this section. And, the result I got is surprisingly beautiful (to me, at least). So, I decided to share it. Here it is:Suppose the number of purge hit, X follows a binomial distribution(N,p) and N follows a Geometric distribution(p), where N = number of total hits, p = proc. rate of purge, and p+q=1.Then, the expected value of X = E[ E(X l N) ] ; and the variance of X = Var[ E(X l N) ] + E[ Var(X l N) ] (look it up online for more info.)Facts:If A~Bin(n,p), then E(A)=np and Var(A)=npq ; If B~Geo(p), then E(B)=1/p and Var(B)=q/(p^2) .Thus:E(X) = E[ E(X l N) ] = p/p = 1Var(X) = Var[ E(X l N) ] + E[ Var(X l N) ] = (p^2)*q/(p^2) + pq/p = q+q = 2qIt turns out that the expected value of X is a constant, 1 ; and the variance of X is 2q or 2*(1-p), which is unexpectedly beautiful. Taking spirit blackhole for example, the proc. rate is 5%. So: E(X) = 1 and Var(X) = 2*0.95 =1.9As you can see, I must be so bored to type out all this craps which I doubt anyone would understand or even bother to read. LMAO. Anyway, if you have made it here, then you are really good.

Glad to hear that you find beauty in it.To me it all sounds pretty normal, nothing exciting.

...I don't understand. But still really cool. Makes me wish I paid attention in high school.

Lol, wanna do the maths exam for me ?