Question

Gary Oak is on a mission to complete his Pokedex before Ash Ketchum. To this end, he startssearching the tall grass for Pokemon. Assume that there aremPokemon in total and that Garyhas seen none of them at the start. Assume also, that all of themPokemon are equally likely toappear in the tall grass and each appearance is independent of the previous appearances. LetKbe the number of encounters required for Gary to fill his Pokedex (A filled Pokedex means that hehas seen allmPokemon atleast once). Find the expectation and variance of K.Hint: Find the expected number of encounters to find a new Pokemon having already seenkPokemon and use this to find the required quantities.

Answer 1

Answer:

Variance Σ i=1 (1 - Pi) /Pi^2

Expected value= π(1/1 + 1/2 .......... 1/n) = π.Hn

Step-by-step explanation:

From the information given, we can get

E(T) = E(t1) + E(t2) ......... +E(tn)

1/P1 + 1/P2.......... 1/Pn

-π/r + π/ r-1 +........... π/1

π(1/1 + 1/2 .......... 1/n)

To find the variance,

Variance= var (x1 + x2......... +xr)

E i=1 var (Xi)

Where,

x1, x2, x3 ........... are all independent of each other.

In case of xi : var (xi) = E(xi^2) - E(xi^2)

Probability of the ith term of coupon that would be observed

Pi = (n - i - 1)/n

Therefore,

i-1 coupons out of a total of n coupons.

To calculate Exi, probability Pi

The expected number of coupons required should be = 1

In probability, 1 - Pi expected number of coupons required E (Xi +1)

E(Xi) = Pi + (1- Pi) E(xi + 1)

E(Xi) = 1/Pi

Due to the above,

E(Xi)^2= Pi + (1- Pi) E(xi + 1)^2

E(Xi)^2= Pi + (1- Pi) E(xi^2 + 1 + 2xi)

E(Xi)^2= 2/Pi^2 - 1/Pi

Var (Xi) = 2/Pi^2 - 1/Pi - 1/Pi^2

= (1 - Pi) /Pi^2

Variance Σ i=1 (1 - Pi) /Pi^2