Math Help - PMF

1. PMF

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

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

(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!

2. I don't know what the star mean, usually it's multiplication, seems to mean such that here.
But this looks like a geometric random variable.

3. Originally Posted by matheagle
I don't know what the star mean, usually it's multiplication, seems to mean such that here.
But this looks like a geometric random variable.
N* = {1,2,....}
I don't understand what you said ><

Anyway,

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

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

(i) Determine a such that P is the probability mass function.
The sum of all possible values of the pmf has to be 1.
So basically, find a such that $\sum_{k=1}^\infty a3^{-k}=a \sum_{k=1}^\infty 3^{-k}=1$ (this is merely a geometric series )
(ii) Is X more likely to take even or odd values?
So let's find out the probability that X has even values.
This is P(there exists k in N* such that X=2k)=P(X=2*1 OR X=2*2 OR X=2*3 OR ...)
This can be written as $\mathbb{P}\left(\bigcup_{k=1}^\infty \{X=2k\}\right)$
And since the events $\{X=2k\}$ are disjoint for different k, this probability is :
$\sum_{k=1}^\infty \mathbb{P}(X=2k)=\sum_{k=1}^\infty a3^{-2k}=a \sum_{k=1}^\infty 9^{-k}$

If you get that this probability is 1/2, then it means that the probability that X take odd values is 1/2 too (do you understand why ?)
And this would mean that X has equal probability to get even or odd values.

(iii) If they exist, calculate E(x) and Var(x).
Do you think they exist ?

Prove or disprove that $a \sum_{k=1}^\infty k3^{-k}$ converges