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Unit 05 /
Problem 3:


Properties of estimators
|
Easy


(Get Paid Back for Shoddy Chips)

Suppose that you buy $n$ silicon chips, numbered $1,2,..,n$. Let random variable $L_i$ represent the lifespan of chip $i$. Assume all $L_i$ are i.i.d. with common expectation 2. The manufacturer agrees to refund you an amount (in dollars) equal to $(4 - \sum_{i=1}^n L_i/n)^2 $.

Prove that as $n \to \infty$, the probability limit of your refund is 4 dollars.

Solution:

We start by noting that we can rewrite the guarantee without changing it:

$$\begin{aligned}(4 - \sum_{i=1}^{n} \frac{L_{i}}{n})^{2} &= (4 - \frac{1}{n} \sum_{i=1}^{n} L_{i})^{2} \\&= (4 - \overline{X})^{2}\end{aligned}$$

We can get this done quite quickly, if we note that WLLN says that $\overline{X} \overset{p}\rightarrow E[X]$.

$$(4 - \overline{X})^{2} \overset{p}\rightarrow (4- E[X])^{2} = (4 - 2)^{2}.$$

To be complete, we should note that the CMT says that if $T_{(n)} \overset{p}\rightarrow c$ that $g(T_{(n)}) \overset{p}\rightarrow g(c)$, and so the statement we are interested in $\overset{p}\rightarrow (4 - 2)^{2} = 4$.