Gary Oak is on a mission to complete his Pokedex before Ash Ketchum. To this end, he starts searching the tall grass for Pokemon. Assume that there are m Pokemon in total and that Gary has seen none of them at the start. Assume also, that all of the m Pokemon are equally likely to appear in the tall grass and each appearance is independent of the previous appearances. Let K be the number of encounters required for Gary to fill his Pokedex (A filled Pokedex means that he has seen all m Pokemon atleast once). Find the expectation and variance of K.

Let denote the number of encounters required to see a new Pokemon after having encountered of them. Then .

If Gary has already seen Pokemon, that leaves still to be encountered, and the next new encounter has a probability of of occurring. Then is geometric, with PMF

Then the expectation and variance of are

is the total number of encounters needed to record all Pokemon, so

By linearity of expectation,

The are independent, so the variance is

why do wages differ across industries

step-by-step explanation :

micro economists are concerned with economics on an industry or individual firm basis.

macro economists are concerned with national and international level