Statistics question

Question

Let X and Y be two discrete random variables with joint probability mass function Pxy (x, y). Show that marginal probability mass function of X can be calculated as follows:

P_x(X) = Σ_y∈_Sy P_xy(x,y)

Your proof must contain sample space description and probability axioms.

Answer 1

Hi Ibrahim! The idea for this proof should probably go something like this. If S_x is the sample space for X and S_y is the sample space for Y, we can denote the space for (x,y) by S_X × S_Y - in other words, this is the set of all pairs of outcomes for x and y. From the definition of a joint probability mass function, we have P_X,Y(x, y) = P(X = x AND Y = y). The idea behind what you should do next is to notice that P_X(x) should encompass all results where X = x. Now we are almost done. The last question is are the events y_j ∈ S_Y mutually exclusive for all j?

Let's see a concrete example. Let X denote the random variable corresponding to the outcome of a dice roll and let Y denote a similar but independent random variable. We would like for P_X(x) = 1/6 for all x where 1 ≤ x ≤ 6. P_X,Y(x,y) = 1/36 for all pairs (x,y) where 1≤x≤6, and 1≤y≤6, which is by independence. We can see that all outcomes y are mutually exclusive, so the probability P_X(x) = P((X = x AND Y = 1) OR (X = x AND Y = 2) OR ...). By the axioms of probability, P((X = x AND Y = 1) OR (X = x AND Y = 2) OR ...) = P(X = x AND Y =1) + P(X=x AND Y = 2) + .... If you substitute in all of the probabilities you will get the result above. You should try doing the same but for the case where X and Y correspond to the outcome of the same dice roll and see that this gives the same result.

I hope this helps!