Question $-6$ points

Consider a camera shop that sells two different models A and B with correlated

demand. The demand for these models is modeled using a bivariate Poisson

distribution. The joint probability function for demands $D_1$ and $D_2($for model

A and model $B$ respectively$)$ is:

$P_{D_1,D_2}(d_1,d_2)=$exp$(-({}_1+{}_2+{}_3))(\frac{{}_1^{d_1}{}_2^{d_2}}{d_1!d_2!}){\displaystyle \underset{j=0}{\overset{\text{m}\text{i}\text{n}(d_1)}{}}},d_2(\frac{d_1}{j})(\frac{d_2}{j})j!(\frac{{}_3}{{}_1{}_2}{)}^j$

The given parameters are ${}_1=1\mathrm{.}5,{}_2=1\mathrm{.}5,$ and ${}_3=0\mathrm{.}5.$

Required

The retailer enforces a policy to maintain a maximum inventory of $5$ units for

each model. The replenishment policy states that whenever the total inven$-$

tory for both models drops to $2$ units or fewer, the inventory for each model

is replenished back to $5$ units by the next day. Assume lost sales, meaning if

demand exceeds available inventory, customers only buy what is available.

Required

Define the state space of the provided system.

construct the state transition matrix for the described system $($answer this

part by providing computer program $+$ its output using Python $($preferred$),$

Matlab or R$.$

Discuss how the correlation between the product demands $($indicated by ${}_3)$

affects inventory levels.