# Law of iterated expectations - an small exercise

Let $X_i$ be the number of widgets in a particular box $i$.
Let $N$ be the number of boxes in a crate.

Assume $X$ and $N$ are independent, both with expected value equal to 10 and variance equal to 16.

Evaluate the expected value of $T$, where $T$ is the total widgets in a crate.

Is it wrong to take the following aproach?

(1) Law of iterated expectations: $$E[T] = E(E[T|N])$$

(2) Since in a crate there will be $X.N$ widgets, with both $X$ and $N$ random, $$E[E(T|N)]= E[N.X]$$

(3) Since $X$ and $N$ are independent, $$E[X.N] = E[X].E[N]=10^2=100$$