From edX MIT probability course - Widgets and Crates:

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, $$E[X.N] = E[X].E[N]=10^2=100$$

Thanks in advance,

  • 1
    $\begingroup$ That's exactly how I'd do it. And you can compute the variance from the corresponding variance law. (Since this is essentially a routine bookwork problem, please add the self-study tag.) $\endgroup$ – Glen_b Jul 8 '14 at 6:46

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