Correlation coefficient?

Question

Can someone help.
If the Marginal Distribution of X is
x 0 1 2 3
P(x) 1/6 2/6 2/6 1/6

AND
The Marginal Distribution function of Y is
y 0 1 2
p(y) 1/4 2/4 1/4

Find the correlation coefficient.

Answer 1

The correlation coefficient ρ_XY is given by

ρ_XY = σ_XY/(σ_X σ_Y)

where σ_XY is the covariance of X and Y, and σ_X and σ_Y are the standard deviations for X and Y respectively.

A useful formula for σ_XY is

σ_XY = E[XY] - E[X] E[Y]

So you need to find the probability function for XY.

Make a table like the one below. The possible x-values are in the first row, and the possible y-values are in the first column. Each of the other table entries is the product of the x-value in its column and the y-value in its row.

.. | 0 | 1 | 2 | 3
--------------------
0 | 0 | 0 | 0 | 0
1 | 0 | 1 | 2 | 3
2 | 0 | 2 | 4 | 6

The probability that the product is 0 is

(1/4)(1/6) + (1/4)(2/6) + (1/4)(2/6) + (1/4)(1/6) + (2/4)(1/6) + (1/4)(1/6) = 3/8

Similarly, you can find that

P(XY=1) = 1/6
P(XY=2) = 1/4
P(XY=3) = 1/12
P(XY=4) = 1/12
P(XY=6) = 1/24

Then

E[XY] = 0*3/8 + 1*1/6 + 2*1/4 + 3*1/12 + 4*1/12 + 6*1/24 = 3/2

E[X] = 0*1/6 + 1*2/6 + 2*2/6 + 3*1/6 = 3/2
E[Y] = 0*1/4 + 1*2/4 + 2*1/4 = 1

Then

E[XY] - E[X] E[Y] = 3/2 - (3/2)(1) = 0

So the covariance σ_XY = 0 and so the correlation coefficient ρ_XY = 0 (no need to compute σ_X and σ_Y).

Answer 2

in accordance to N.Shaviv & R. Pielke the contributions of AGHG to climate replace is approximately 20% and something is as a results of organic forcings. Correlations suggestions-set has many weak spot as mentioned by ability of R. Grumbine. One final piece. solid your eyes decrease back as much as the graph, and consistent with danger click on it to get the whole length version. in direction of the suitable hand end, you notice a dot this is a techniques above the right this moment line extra healthful. you at the instant are not shocked that this is 1998 -- the 365 days of the main nicely known El-Nino that turn into concurrent with a time of intense photograph voltaic interest. The bit farthest under the curve around 357 ppm CO2 is 1992-3 -- cooling effects of Mount Pinatubo. Even against a statistically solid trend, there are nevertheless different effects in the device that could supply some tenths of a diploma climate modifications. there are a number of reasons climatologists do not suggestions-set climate this way, and that they are solid ones. I basically did it because of the fact i've got been encountering components that say that the correlation is 0 (nonexistent). those places are incorrect.