Problem
I have a random variable vector (X,Y) whose joint mass function is given by the below table (X = j, Y = k).
Additionally, the correlation of the vector is given by : E[XY] = probability of X = Y (so s + t according to the table). s + t = 0.62
Let W = g(X) where g(X) is the linear estimation of Y with respect to X. So W = g(X) = aX + b.
Determine the mass function of W.
What I tried so far :
Determine the covariant:
$ Cov = E(XY) - E(X)E(Y)$
$ E(X) = 1*(0.07+t) + 2*(0.13+0.12) = 0.57 + t $
$ E(Y) = 0.18 + t $
$ ==> Cov = t^2 + 0.75t - 0.52$
Find alpha
$a = \frac{COV(X,Y)}{VAR(X)}$ with $VAR(X) = E(X^2) - E(X)^2$ (best linear predictor)
My issue
By resolving the above equations, I still cannot get rid of the "t" variable. I don't know how I'm supposed to find a real number for "a" and "b" if I don't have the values for s and t, unless I missed something.
Any help would be appreciated.