Question

Let X1, X2 ... be a sequence of random variables. Write down the definition of "Xin con- verges to X in distribution". State the central limit theorem formally. Make sure to include all the assumptions of the theorem. Ten people wait in line to pay their items at a store. Suppose the amount of time (in minutes) that each customer takes to pay is a Uniform([1, 3]) random variable. Use the central limit theorem to approximate the probability that it will take less than 22 minutes in total for the 10 customers to pay for their items.

Answer 1

4) (a) X₁ , X₂ , ..... sequence of RVs

X_n converges to X in distribution :

Notation : X_n  $\overset{D}{\rightarrow}$ X

Definition : Consider a sequence {X_n} of RVs defined on probability space $(\Omega , A, P)$ , with corresponding sequence of DFs
Fnac
. The sequence {X_n} of RVs converges in distribution/law to X , with distribution fⁿ F , if  $F_{n}(x) \rightarrow F(x)$ as n
1 00
, at every continuity points of F(x).

(b) Central Limit Theorem (by Lindeberg-levy): Let X₁, ...X_n,... be a sequence of independent and identical RVs. $\bar{X_{n}} = \frac{1}{n}\sum_{1=1}^{n} X_{i}$

Then, $\frac{\sqrt{n}(\bar{X_{n}-\mu})}{\sigma } \sim ^{a} N(0,1) as n \rightarrow \infty$

where, $\mu = E(X_{i}) ,\forall i$ & $\sigma ^{2} = V(X_{i}) \forall i$

(c) Let X = amount of time (in minutes) that each customer takes to pay

here, X$\sim U([1,3])$

$\therefore \mu = E(X) = \frac{1+3}{2}= 2$

$\sigma ^{2} = V(X)=\frac{(3-1)^{2}}{12} = \frac{1}{3}$

So, by CLT, $\frac{\sqrt{n(\bar{X}-\mu)}}{\sigma } \sim ^{a} N(0,1)$ where X_i$\sim ^{i\phi } U ([1,3])$

Now probability that it will take less than 22 minutes in total for 10 customers to pay for their lines

= $P\left [ \sum_{i=1}^{10}X_{i} < 22 \right ] = P\left [ \bar{X}< 2.2 \right ]$

= $P(\frac{\sqrt{10}(\bar{X}-2)}{\frac{1}{3}}<\frac{\sqrt{10}(2.2-2)}{\frac{1}{3}})$

= $P\left [ Z < 1.897 \right ]$ , where Z $\sim N(0,1)$

= $\phi (1.897)= 0.9706$