Question

Joint mass functions

Question will follow

Answer 1

Suppose x and y are independent variable with f(x)=.8, .15, and .05 for x=0, 1, 2; and f(y)=.5, .25, .15, .08, .02 for y=0,1,2,3,4.

You do you get the joint mass function of this?

Answer 2

If \(X, Y\) are independent, then the joint mass function \(P(X=x, Y=y)\) is:
\[ P(X=x, Y=y)=P(X=x)P(Y=y)\]

That is for every (x,y) pair, you must multiply each value from the probability density functions.

say you have X=0, Y=2
Then P(X=0, Y=2) = P(X=0)P(Y=2) = 0.8*0.15 = 0.12

A nice way to do it might be in a table like this: I filled up the first 3 values in the first column to give you an idea.

Answer 3

If you prefer, to be consistent with your notation, you can say the joint mass function is \(f(x,y)\), and so you have:
\(f(x,y)=f_X(x)f_Y(y)\)

if X=0, Y=2, then
\(f(0,2)=f_X(0)f_Y(2)=0.8*0.15=0.12\)

Answer 4

Is it possible to write a function, such as f(x,y)=2x-y (just an example)?

Answer 5

no not really, you are given the mass functions are a discrete random variable with no apparent function associated with it (and that's ok, not all probability mass functions are defined by a formula)

Answer 6

you might have seen mass functions given before in a table.. that's because there is no formula associated with it, like say: