# p.m.f.

• May 12th 2009, 06:54 AM
redpack
p.m.f.
Let x and y have the joint p.m.f. given by:

$\displaystyle \begin{pmatrix}&&&x&&\\y&1&2&3&4&5\\3&0.02&0.03&0. 06&0.15&0.07\\2&0.02&0.05&0.08&0.12&0.06\\1&0.05&0 .05&0.06&0.13&0.05\end{pmatrix}$

a) Find the marginal p.m.f.s and the means and variances.
b) Are x and y independent? Why?
c) Find the covariance and the correlation coefficient of x and y.

I appreciate any help. I don't know how a problem like this is done so can't do the rest of them. Thank you so much
• May 12th 2009, 05:06 PM
matheagle
SUM across to get the values, that is where the name marginal comes from.

The $\displaystyle P(Y=3)=.02 + .03 + .06 + .15 + .07$
Do that across and down (for x's distribution).
IN order for these two rvs to be independent... for EVERY position the sum across and the sum down's product must equal the probability of that position.

The shortcut formula for covariance is $\displaystyle Cov(X,Y)=E(XY)-E(X)E(Y)$ and if the rvs are indep, this is zero.

The definition of correlation is $\displaystyle {Cov(X,Y)\over \sigma_x\sigma_y}$ and if the rvs are indep, this is zero.
• May 12th 2009, 06:07 PM
redpack
Thank you!

I am also supposed to:
a) sketch the support of X and Y
b) Find the best-fitting line and draw it on the figure

I have no examples so I don't know what is being asked for. Do you have any ideas of what they're looking for?
• May 12th 2009, 09:52 PM
matheagle
the support are just the points that have positive probability...
(1,1),..., (5,1)
(1,2),..., (5,2)
(1,3),..., (5,3).
I'm not sure what is meant by the best fitting line.
I can plot a least squares line through these 15 points.
Are we supposed to weight these points via the probabilities?
• May 15th 2009, 09:37 PM
redpack
I'm sorry, this question is still killing me! So I just sum across to get 3 marginal p.m.f.s for y and sum down to get 5 p.m.f.s for x? Then how would I calculate the means and variances for this type of problem?

"IN order for these two rvs to be independent... for EVERY position the sum across and the sum down's product must equal the probability of that position."
My probability is just out of 15? For example 0.05 occurs 4 times so P(0.05)=4/15?

I'm totally lost on the covariance and correlation coefficient.

For the graph, I simply graphed the 15 points.

Thank you for everything. I'm sorry I am having such a hard time grasping this concept.