random variable X,
random variable Y1,Y2,Y3...Yn are independent identically distributed
we have known P(X=1|Y=y)=f(y);
consider
P(X=1|Y1=y1)=f(y1), P(X=1|Y2=y2)=f(y2),......P(X=1|Yn=yn)=f(yn),
will f(y1),f(y2),..,f(yn) be the same?
will f(y2)* f(y2)* f(yn)=f(y)n?
To me this is just a matter of notations. If fi(y)=P(X=1|Yi=y) then for any i and j you have fi(y)=fj(y). So obviously, f1(y)*...*fn(y)=f(y)n
Yeah, my first question would be what do you mean by "f(y1)* f(y2)* f(yn)=f(y)^n"? Given your notation Y is random variable that takes on values from, at the most general level, a countable set {y1,y2,....}, For simplicity say it's discrete and finite, then the left side of the equation makes sense, and must be interpreted as a product of values, but then why does the right side even mean??? For the first responder, note the difference between the placement of subscripts in your explanation versus the given question. If I was presented with the final equation and told to interpret it out of context, all it says is that f is a constant function... I'm a bit too tired to address the stuff you wrote before that (I'm sleepy) but in short, I do not think you are correct.
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