本题目来源于试卷: IB MAI HL Statistics & Probability Topic 4.3 Probability，类别为 IB数学
[问答题]
Two grocery stores, storeur1lywt, cp xs2s;6qg mb, k4dl18sy8 A aneyc)1e+wi-hy(fwn 0uu8ztp(q1;bkk d store B, serve in a small city. Each year, store A keeps 30 % of its cus0c1py;q( n- kwifub8y (k1we+h )uetztomers while 70 % of them switch to store Store B keeps 60 % of its customers while 40 % of them switch to store A.
1. Write down a transition matrix $\boldsymbol{T}$ representing the proportions of the customers moving between the two stores.
At the end of 2019 , store A had 8360 customers while store B had 6820 customers.
2. Find the distribution of the customers between the two stores after two years.
3. 1. Show that the eigenvalues of $\boldsymbol{T}$ are $\lambda_{1}=1$ and $\lambda_{2}=-0.1$ .
2. Find a corresponding eigenvector for each eigenvalue from part (c) (i).
3. Hence express $\boldsymbol{T}$ in the form $\boldsymbol{T}=\boldsymbol{P} \boldsymbol{D} \boldsymbol{P}^{-1}$ .
4. Show that
$\boldsymbol{T}^{n}=\frac{1}{11}\left[\begin{array}{ll}
4+7(-0.1)^{n} & 4-4(-0.1)^{n} \\
7-7(-0.1)^{n} & 7+4(-0.1)^{n}
\end{array}\right]$
, where n $\in \mathbb{Z}^{+}$ .
5. Hence find an expression for the number of customers buying groceries from store $\mathrm{A}$ after n years, where n $\in \mathbb{Z}^{+}$
6. Verify your formula by finding the number of customers buying groceries from store A after two years and comparing with the value found in part (b).
7. Write down the long-term number of customers buying groceries from store A.
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