Mathematics, 24.04.2020 17:49 lwattsstudent
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.
Answers: 2
denote the number of encounters required to see a new Pokemon after having encountered 
of them. Then 
.
still to be encountered, and the next new encounter has a probability of 
of occurring. Then 
![E[K_i]=\dfrac1{p_i}=\dfrac m{m-(i-1)}](/tpl/images/0624/3107/910e5.png)
![V[K_i]=\dfrac{1-p_i}{{p_i}^2}=\dfrac{m(i-1)}{(m-(i-1))^2}](/tpl/images/0624/3107/00b7e.png)
is the total number of encounters needed to record all 
Pokemon, so
![E[K]=\displaystyle\sum_{i=1}^mE[K_i]=\sum_{i=1}^m\frac m{m-(i-1)}=\frac mm+\frac m{m-1}+\frac m{m-2}+\cdots+\frac m1](/tpl/images/0624/3107/6099c.png)
![E[K]=m\displaystyle\sum_{i=1}^m\frac1i](/tpl/images/0624/3107/a729d.png)
![V[K]=\displaystyle\sum_{i=1}^mV[K_i]=\sum_{i=1}^m\frac{m(i-1)}{(m-(i-1))^2}](/tpl/images/0624/3107/79116.png)
![V[K]=m\displaystyle\sum_{i=0}^{m-1}\frac i{(m-i)^2}](/tpl/images/0624/3107/1428f.png)
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Gary Oak is on a mission to complete his Pokedex before Ash Ketchum. To this end, he starts searchin...
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