STEM Problem Detail

Problem Statement

Prompt

GIVE THE SOLUTION OF EXAMPLE 1.7

Solution Rendering...

**Solution**

We are given that
\[
G(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-u^{2}/2}\,du,
\]
where $$G(x)$$ is the distribution function of the standard normal distribution.

We want
\[
G(x)=P(Z>x),
\]
where $$Z\sim N(0,1)$$.

Let $$\bar{X}$$ be the mean of a sample of size $$n$$ from a population with mean $$\mu$$ and standard deviation $$\sigma$$.
Then $$\bar{X}$$ is normally distributed if the population is normal, and is asymptotically normal by the Central Limit Theorem otherwise, with
\[
E(\bar{X})=\mu, \qquad \operatorname{Var}(\bar{X})=\frac{\sigma^{2}}{n}.
\]

After a shift of the mean by $$k\sigma$$, the distribution becomes
\[
\bar{X}\sim N\!\left(\mu+k\sigma,\;\frac{\sigma^{2}}{n}\right).
\]

Hence,
\[
Z=\frac{\bar{X}-(\mu+k\sigma)}{\sigma/\sqrt{n}}\sim N(0,1).
\]

Now,
\[
P_n=P(\bar{X}>UCL_{\bar{X}})
=P\!\left(\bar{X}>\mu+3\frac{\sigma}{\sqrt{n}}\right).
\]

Therefore,
\[
P_n
=P\!\left(Z>\frac{\mu+3\frac{\sigma}{\sqrt{n}}-(\mu+k\sigma)}{\sigma/\sqrt{n}}\right)
=P(Z>3-k\sqrt{n}).
\]

Thus,
\[
P_n=G(3-k\sqrt{n}).
\]

Let $$\bar{X}_i$$ denote the mean of the $$i$$th sample of size $$n$$. Then
\[
p_n=P(\bar{X}_i>UCL_{\bar{X}}), \qquad
q_n=P(\bar{X}_i\le UCL_{\bar{X}})=1-p_n.
\]

If $$Y$$ denotes the number of samples required for the first sample mean to exceed $$UCL_{\bar{X}}$$, then $$Y$$ follows a geometric distribution with parameter $$p_n$$ and probability function
\[
P(Y=r)=q_n^{\,r-1}p_n,\qquad r=1,2,\ldots
\]

Hence,
\[
E(Y)=\sum_{r=1}^{\infty} r\,q_n^{\,r-1}p_n
=p_n(1+2q_n+3q_n^2+\cdots).
\]

Let
\[
S=1+2q_n+3q_n^2+4q_n^3+\cdots .
\]

Then,
\[
q_n S=q_n+2q_n^2+3q_n^3+\cdots,
\]
and subtracting,
\[
(1-q_n)S=1+q_n+q_n^2+\cdots=\frac{1}{1-q_n}.
\]

Hence,
\[
S=\frac{1}{(1-q_n)^2}=\frac{1}{p_n^2}.
\]

Substituting in the expression for $$E(Y)$$,
\[
E(Y)=p_n S=\frac{p_n}{p_n^2}=\frac{1}{p_n}.
\]

Thus, the expected number of samples required for the first point to exceed the control limit is
\[
E(Y)=\frac{1}{p_n}.
\]

Loading solution...

Problem #86