Exercise 3 — Candidates who have followed the specializationEach young parent uses only one brand of baby food jars each month. Three brands $\mathrm { X } , \mathrm { Y }$ and Z share the market. Let $n$ be a natural integer. We denote: $\quad X _ { n }$ the event ``brand X is used in month $n$ '', $Y _ { n }$ the event ``brand Y is used in month $n$ '', $Z _ { n }$ the event ``brand Z is used in month $n$ ''. The probabilities of events $X _ { n } , Y _ { n } , Z _ { n }$ are denoted respectively $x _ { n } , y _ { n } , z _ { n }$. The advertising campaign of each brand causes the distribution to change.
A buyer of brand X in month $n$ has the following month: $50 \%$ chance of remaining loyal to this brand, $40 \%$ chance of buying brand Y, $10 \%$ chance of buying brand $Z$.
A buyer of brand Y in month $n$ has the following month: $30 \%$ chance of remaining loyal to this brand, $50 \%$ chance of buying brand X, $20 \%$ chance of buying brand $Z$.
A buyer of brand Z in month $n$ has the following month: $70 \%$ chance of remaining loyal to this brand, $10 \%$ chance of buying brand X, $20 \%$ chance of buying brand Y.
- a. Express $x _ { n + 1 }$ as a function of $x _ { n } , y _ { n }$ and $z _ { n }$.
We admit that: $y _ { n + 1 } = 0.4 x _ { n } + 0.3 y _ { n } + 0.2 z _ { n }$ and that $z _ { n + 1 } = 0.1 x _ { n } + 0.2 y _ { n } + 0.7 z _ { n }$. b. Express $z _ { n }$ as a function of $x _ { n }$ and $y _ { n }$. Deduce the expression of $x _ { n + 1 }$ and $y _ { n + 1 }$ as functions of $x _ { n }$ and $y _ { n }$.
2. We define the sequence $\left( U _ { n } \right)$ by $U _ { n } = \binom { x _ { n } } { y _ { n } }$ for every natural integer $n$.
We admit that, for every natural integer $n$, $U _ { n + 1 } = A \times U _ { n } + B$ where $A = \left( \begin{array} { l l } 0.4 & 0.4 \\ 0.2 & 0.1 \end{array} \right)$ and $B = \binom { 0.1 } { 0.2 }$.
At the beginning of the statistical study (January 2014: $n = 0$), we estimate that $U _ { 0 } = \binom { 0.5 } { 0.3 }$. Consider the following algorithm:
| Variables | \begin{tabular}{l} $n$ and $i$ natural integers. |
| $A$, $B$ and $U$ matrices |
\hline Input and initialization &
| Request the value of $n$ $i$ takes the value 0 |
| $A$ takes the value $\left( \begin{array} { l l } 0.4 & 0.4 \\ 0.2 & 0.1 \end{array} \right)$ |
| $B$ takes the value $\binom { 0.1 } { 0.2 }$ |
| $U$ takes the value $\binom { 0.5 } { 0.3 }$ |
\hline Processing &
| While $i < n$ |
| $U$ takes the value $A \times U + B$ |
| $i$ takes the value $i + 1$ |
| End while |
\hline Output & Display $U$ \hline \end{tabular}
a. Give the results displayed by this algorithm for $n = 1$ then for $n = 3$. b. What is the probability of using brand X in April?
In the rest of the exercise, we seek to determine an expression of $U _ { n }$ as a function of $n$. We denote by $I$ the matrix $\left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $N$ the matrix $I - A$.
3. We denote by $C$ a column matrix with two rows. a. Prove that $C = A \times C + B$ is equivalent to $N \times C = B$. b. We admit that $N$ is an invertible matrix and that $N ^ { - 1 } = \left( \begin{array} { l l } \frac { 45 } { 23 } & \frac { 20 } { 23 } \\ \frac { 10 } { 23 } & \frac { 30 } { 23 } \end{array} \right)$.
Deduce that $C = \binom { \frac { 17 } { 46 } } { \frac { 7 } { 23 } }$.
4. We denote by $V _ { n }$ the matrix such that $V _ { n } = U _ { n } - C$ for every natural integer $n$. a. Show that, for every natural integer $n$, $V _ { n + 1 } = A \times V _ { n }$. b. We admit that $U _ { n } = A ^ { n } \times \left( U _ { 0 } - C \right) + C$.
What are the probabilities of using brands $\mathrm { X } , \mathrm { Y }$ and Z in May?