Question

Widgets are placed into boxes, boxes are packed into crates, and a collection of crates constitutes a shipment. Consider a non-negative, integer-valued random variable U, with expectation equal to 10, and variance equal to 16. Independent experimental values of this random
variable describe:
• X, the number of widgets in any particular box.
• N, the number of boxes in any particular crate.
• K, the number of crates in any particular shipment.
Evaluate the expectation and variance for:
(a) T, the number of widgets in any particular crate.
(b) W, the total number of widgets in any particular shipment.

Answer 1

1. Let X_I be the number of widgets in the ith box, so Xi~X.

We are given that there are N boxes in a crate.

Thus, the number of widgets in a crate is

Now, Using Wald’s Equation, the expected value is:

Now, Using Variance formula, we get

b) Let T_I be the number of widgets in the ith crate, so Ti~T from part(a).

We are given that K crates in a particular shipment.

So,

Now by Wald’s Equation, the expected value is:

Now, Using Variance formula, we get

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