Question

Two balanced dice are rolled. Let X be the sum of the two dice. a) Obtain the probability distribution of X. b) Find the mean and standard deviation of X.

Answer 1

Try finding all the possible sums you can get from two dice.
For example:
To get the sum "2", you need the dice to show (1,1)
To get the sum "4", you get get either (1,3), (2,2) or (3, 1)

Try this for all sums from 2 to 12 (since the minimum sum you get get is 2, and the max is 12).

Then to find the probability of each event (each "sum") will just be the number of outcomes for each sum over the number of all possible outcomes in the sample space.

That is,
\[ P(sum~is~"x")=P(X=x)=\frac{number~of~ways~to~get~x}{total~number~of~all~possible~outcomes}\]

Then, \[mean=E(X)=\sum_{all~x}x\cdot P(X=x)\]
\[ std~deviation=\sqrt{Var(X)}=\sqrt{E(X^2)-[E(X)]^2}\] where \[E(X^2)=\sum_{all~x}x^2\cdot P(X=x) \]