1. Let X be a random variable in N* = {1,2,....} such that

$\displaystyle \forall k \in N^{*} P (X=k) = a3^{-k}$

(i) Determine a such that P is the probability mass function.

(ii) Is X more likely to take even or odd values?

(iii) If they exist, calculate E(x) and Var(x).

2. Ten phones are linked by only one line to the network. Each phone needs to log on on average 12mm an hour to the network. Calls made by different phones are independent of each other. They can't call simultaneously.

(i) What is the probability that k phones (k = 0,1, ... , 10) simultaneously need the line? What is the most likely number of phones requiring the line at one time?

(ii) We add lines to the network, allowing simultaneous calls. What is the minimum number c of lines to have so that on average there is no call overloading more than 7 times out of 1000?

Thank you!