Answer to Question #144462 in Statistics and Probability for edoCRetae

Question

4. Each member of a group of n players rolls a die.
(a) For any pair of players who throw the same number, the group scores 1 point.
Find the mean and variance of the total score of the group.
(b) Find the mean and variance of the total score if any pair of players who throw
the same number scores that number.
5. Every package of some intrinsically dull commodity includes a small and exciting plastic
object. There are k different types of object, and each package is equally likely to
contain any given type. You buy one package each day.
(a) Find the mean number of days which elapse between the acquisitions of the n
th
new type of object and the (n + 1)th new type.
(b) Find the mean number of days which elapse before you have a full set of k objects.

Answer 1

4)a) Let S_i denote the score obtained by the players who throw i,i=1,2,…,6 , and let X_i be the number of people who throw i, and 1_ij  be the indicator function which is 1 only if the j^th person throws i. So, we have X_i=∑ⁿ_j=11_ij.  E[S_i|X_i]=X_i(X_i−1)/2 and so E[S_i]=(n²−n)/72 and therefore E[S]=(n²−n)/12, where S=∑⁶_i=1S_i is the total score.

b) The indicator variables Y_ij that are 1 if i and j throw the same number are pairwise independent and can thus be used to find both the expectation and the variance. The expectation of Y_ij is 1/6 and the variance is $\frac16\cdot\left(1-\frac16\right)=\frac5{36}$ ,so the expectation of the score is $\binom n2\cdot\frac16=\frac{n(n-1)}{12}$ and the variance of the score is $\binom n2\cdot\frac5{36}=\frac{5n(n-1)}{72}$

5)suppose there are 6 types of objects

the probabilities of getting a new type go on decreasing from 1 to 5/6, to 4/6....as we get more & more types, and using E[x] = 1/p, the expected # of days to get the next type goes on increasing

E[days to get 1st type] = 6/6 ...... 6/(7-1)

E[more days to get 2nd type] = 6/5 ..... 6/(7-2)

E[more days to get 3rd type] = 6/4 ..... 6/(7-3)

a. E[more days to get (j+1)th new object] = c/(c-j)

b. E[days to get full set] = Σ(c-j+1)