you would just have to show that the other 3 hold as well.
You have
and you will need
and they must sum to one via the total and the marginals, so that should help.
I've found a good answer to a particular question I was given, though I'd like a second opinion on whether or not my answer is solid (I might be "begging the question" in this).
Show that two Bernoulli random variables and are independent if and only if .
Here's my answer.
Two Bernoulli random variables are independent if and only if they are uncorrelated, and thus have a covariance of zero ( ).
Let be the pmf of , and let be the pmf of .
If and are independent then, by definition,
,
as .
If on the other hand we have that , then
.
Therefore, and are independent.
Are there any mess-ups I might have made somewhere? I got this answer from this pdf file.
Or, as an alternative, one could do it for where (this would be a bit faster, yet still means the same thing). Note that I listed the question word-for-word, and the question doesn't say anything about marginal pmfs or totals. Though I assume one would want to show that the sum of the marginals equals 1 for completeness, I don't know if such would be necessary for the purpose of the question.
See, I got my answer from this link here, and the proof it gave looks very solid. Is this source missing some necessary parts?
EDIT: This had a doublepost, oops.
But as long as it's here, I figured out what you mean by that "summing" part. Here's what I got from PhysicsForum.
Using this, I'd go into the stuff I put in earlier and be able to complete the proof.
If there's anything I've missed, let me know.