Probability Distribution verification

Question

a probability distribution must satisfy two conditions

1- the probability of each value of the random variable is between 0 and 1.

2- the sum over all the probabilities is equal to 1.

I think to exclude all the zero's mass probability, I should take the summation over all the support of x,y,z.

Can anyone please help me in how to write the verification. I know the concept but donot know how to write it.

Thanks

Answer 1

Since $p(x,y,z) \geq 0$ by definition the only thing you have to prove is that $\sum_{x,y,z} p(x,y,z)=1$. First take the sum over $x$. Since $\sum_x \frac {p(x,y)} {p(y)}=1$ we are left with the proof of $\sum_{y,z} p(y,z)=1$ which is clear.