expected value problem

Question

Say we have a probability density function $f_y(y)=3y^2$, where $0\leq y\leq1$ and we take 15 observations at random. If $x$ is a number within the interval $(.5, 1)$ what is $E(x)$?

Answer 1

I am guessing that you are actually asking for the expectation of $Y$ given that it has this distribution and that it is in the interval $(0.5,1)$.

If so, you should look at $$\frac{\int_{0.5}^1 y f_y(y) \, dy }{\int_{0.5}^1 f_y(y) \, dy }$$ which I am sure you can calculate.

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Or perhaps you want to know the expected number of your sample of fifteen which are in $(0.5,1)$.

In that case, you should look at the probability that any particular one one of them is in that interval $$\int_{0.5}^1 f_y(y) \, dy $$ and then by linearity of expectations, multiply by the number in the sample.