Question

Assume you roll a fair dice twice. Two rolls are independent and identically distributed, with probability of rolling a particular number being 1/6. So, for instance, the probability of rolling 5 and then 2 is P(5,2) = P(5) ⋅ P(2) = 1/6 ⋅ 1/6 = 1/36 Consider a random variable X equal to the sum of two numbers rolled.

what is the mean of X?

In the same setup, find FX(7)

What is the variance of X?

Answer 1

Set up a table of outcomes and the corresponding probabilities to create the probability distribution.
For example, the lowest outcome is
X=2 (=1+1) with probability 1/36.
then
X=3 (1+2 or 2+1) with probability 2/36
...
X=12 (=6+6) with probability 1/36.

Find the mean using
E(X)=μ=Σ(X*P(X)) summed over the sample space.
and variance
σ²=Σ((X-μ)² *P(X)) again summed over sample space.