Verify the joint probability function I had a question I was hoping for some help on:
There are 8 similar chips in a bowl: 3 marked (0;0), 2 marked (1;0), 2 marked (0;1), and 1 marked (1;1). A player selects a chip at random and is given the sum of the two coordinates in dollars.Let $Y_1$ and $Y_2$ represent those two coordinates, respectively.
a) Verify the joint probability function is:
$$f(x,y) = \begin{cases} \dfrac{3-y_1-y_2)}{8} & \text{if $y_1 = 0,1; y_2 = 0,1$} \\ 0 & \text{elsewhere} \end{cases}$$
b) Find E($Y_1 + Y_2$)
c) Find V($Y_1 + Y_2$)

To me it looks like its a discrete probability function, but I'm lost on how to approach a) if that's the case. In addition, if it is the case that it is indeed a discrete probability function, how do you approach finding the expected value in b). I think was confuses me the most is the introduction - That is how (if at all) does the information given to you factor into this question? Would someone be able to help me? Thank you so much in advance for your help! I really do appreciate it!
 A: For each possible combination of y1 and y2, calculate f(y1,y2).  Count up the number of chips with those y1,y2 numbers and show that the probability of getting one of those chips is f(y1,y2).  Example: For y1=0, y2=0, f(0,0)=3/8.  There are 3 chips with (0,0), so the odds of getting one of those is 3/8.  That matches f(0,0).
A: Selecting one chip at random generally means each chip has an equal chance to be chosen. There are $8$ chips, so you can determine $f(x,y)$ by counting the number of chips marked $(x;y)$ and dividing by $8$.
For parts (b) and (c), you could use the joint distribution to compute the marginal expected value and variance of $Y_1$ and $Y_2$, respectively, then use the formulas for the expected value and variance of a sum of random variables. But we were initially given $8$ chips with the following properties:


*

*$3$ chips for which $Y_1 + Y_2 = 0$

*$4$ chips for which $Y_1 + Y_2 = 1$

*$1$ chips for which $Y_1 + Y_2 = 2$


It seems to me this is enough information about the distribution of $Y_1 + Y_2$
to compute the expected value and variance.
I just hope that doesn't defeat the purpose of this exercise.
