Changes to the expected value Calculation

[sushmit86]

Proposing change
.. math:: E[Y^{*}] = P(Y | W=1) - P(Y| W=0),
Since Y is a binary Random variable with possible it would be more appropriate to change it to

.. math:: E[Y^{*}] = P(Y =1 | W=1) - P(Y =1 | W=0),

[sushmit86]

Also IMO it might be nice to add the derivation of the above using the Law of Total Expectation and the fact that E(Y|W=1) = P (Y=1|W=1) Using the fact that for Indicator Random Variables E(Y) = P(Y=1)

[shaddyab]

In theory, Y can also be continuous.

[sushmit86]

if Y is continuous how does the derivation work

E[Y^{*}] = P(Y | W=1) - P(Y| W=0)

What I have is this

E[Y_star] = E (Y * (W - p)/p * (1-p))
= E(Y * (1-p)/p*(1-p)| W=1)* P(W=1) + E(Y * (0 -p)/p(1-p)|W=0) * P(W=0) using Law of Total expectation
Simplifying the above we have

= E(Y|W=1) - E(Y|W=0)
if Y is continuous how are we arriving at the derived result