# Joint p.m.f

• Dec 5th 2009, 01:07 PM
Cickle
Joint p.m.f
I am studying for a final exam and I am having trouble with getting started with one of my problems. If someone could help explain I would be grateful.

Let Y be a random variable taking the values -1,0,1 equally likely. Let V be another random variable, independent of Y, and taking the values +1 & 1 with probability 1/2.

1) Denote by h(y,v) the joint distribution of Y and V. Find h(y,v) (make a table of values)

2) Let X=YV. Find the p.m.f for X

3) Let f(x,y) be the joint p.m.f. of X and Y. Find f(x,y)

4) Are X and Y independent? Reason?

I have looked in my book and I can't seem to find an example of how to construct the table of values. Any help would be appreciated.
• Dec 5th 2009, 01:14 PM
mr fantastic
Quote:

Originally Posted by Cickle
I am studying for a final exam and I am having trouble with getting started with one of my problems. If someone could help explain I would be grateful.

Let Y be a random variable taking the values -1,0,1 equally likely. Let V be another random variable, independent of Y, and taking the values +1 & 1 with probability 1/2.

1) Denote by h(y,v) the joint distribution of Y and V. Find h(y,v) (make a table of values)

2) Let X=YV. Find the p.m.f for X

3) Let f(x,y) be the joint p.m.f. of X and Y. Find f(x,y)

4) Are X and Y independent? Reason?

I have looked in my book and I can't seem to find an example of how to construct the table of values. Any help would be appreciated.

Construct a grid with Y along the top and V along the side. The cells contain the probability of things like Pr(Y = 1, V = -1) etc. and you should know how to calculate these probabilities, especially since the events are independent.

• Dec 5th 2009, 01:47 PM
Cickle
Ok so this is what I am thinking...Is this correct?
• Dec 5th 2009, 01:50 PM
mr fantastic
Quote:

Originally Posted by Cickle
Ok so this is what I am thinking...Is this correct?

Yes.
• Dec 5th 2009, 02:00 PM
Cickle
Thank you so for the second question X=YV find the p.m.f of X

X would take the values of -1,0,1 correct?

and from that how would I find the p.m.f for X
• Dec 5th 2009, 02:01 PM
mr fantastic
Quote:

Originally Posted by Cickle
Thank you so for the second question X=YV find the p.m.f of X

X would take the values of -1,0,1 correct?

and from that how would I find the p.m.f for X

Yes.

Use the table. What cells give you a product of -1? Add up the probabilities. etc.
• Dec 5th 2009, 02:09 PM
Cickle
I guess I am still a little confused so if X=-1 then the it would be 1/6 +1/6 if X=0 it would be 1/6 + 1/6 and if X=1 it would be 1/6?
• Dec 5th 2009, 02:12 PM
mr fantastic
Quote:

Originally Posted by Cickle
I guess I am still a little confused so

if X=-1 then the it would be 1/6 +1/6 Mr F says: Yes.

if X=0 it would be 1/6 + 1/6 Mr F says: Yes.

and if X=1 it would be 1/6? Mr F says: No.

Does 1/3 + 1/3 + 1/6 = 1? So clearly one of these three is wrong ....

Note that 1 = (1)(1) and (-1)(-1).
• Dec 5th 2009, 02:21 PM
Cickle
Sorry I meant 1/6 + 1/6 for the last one too. So how would I write that for it to make sense? f(x)=2/6 for x=-1,0,1 ?

and then for the third question the joint p.m.f. f(x,y)=1/3 * 2/6=1/9 for x=-1,0,1 and y=-1,0,1

and the the fourth they would be independent because f(x,y)=f(x)f(y)

And thank you for your patience I am just trying to make sure that I fully grasp this for my final.