I cannot understand your exponent.
It looks like absolute values, but I doubt you want that.
I bet you mean when .
f(y| )= , y≥ where theta is an unknown positive constant.
Find a moment estimator for theta and adjust theta hat to find an unbiased estimator for theta.
Solving for E(Y) I get which is my sample average (Y-bar).
Am I on the right track?
This is getting weird.
What's x now???
ye^(-y+x) + e^(-theta + x) does not make sense.
I BET you'e "integrating" x times f(y).
THERE is only ONE variable.
Either use x or y, it doesn't matter, these are dummy variables.
JUST integrate yf(y) from theta to infinity.
When you integrate y and plug in bounds, the y is no longer there.
And this x is purely fictional.
you need to review calculus.
You should have ONLY a function of theta as the expectation of Y and you set that to the sample mean.
separate the exponent e^theta and e^(-y) and PULL that outside the integral
then integrate ye^(-y) via parts and plug in theta and let y go to infinity for your bounds.
WHEN you're done there can't be any more y's, only theta's.
Then set that equal to the sample mean.
e^(-theta + theta) =e^0=1