$4.$A$.1$ Two football teams will participate in an international tournament. The estimated probabilities for the numbers of goals these teams will score are shown in the following table:

$\backslash$table$[[,$number of goals scored by team $2],[,0,1,2,3,4,5],[$number of$,0,0\mathrm{.}098,0\mathrm{.}043,0\mathrm{.}035,0\mathrm{.}012,0\mathrm{.}007,0\mathrm{.}003],[$goals scored,$1,0\mathrm{.}142,0\mathrm{.}130,0\mathrm{.}028,0\mathrm{.}008,0\mathrm{.}008,0\mathrm{.}002],[$by team $1,2,0\mathrm{.}094,0\mathrm{.}118,0\mathrm{.}063,0\mathrm{.}008,0\mathrm{.}004,0\mathrm{.}000],[,3,0\mathrm{.}067,0\mathrm{.}043,0\mathrm{.}028,0\mathrm{.}008,0\mathrm{.}004,0\mathrm{.}000],[,4,0\mathrm{.}016,0\mathrm{.}008,0\mathrm{.}004,0\mathrm{.}000,0\mathrm{.}000,0\mathrm{.}000],[,5,0\mathrm{.}016,0\mathrm{.}000,0\mathrm{.}003,0\mathrm{.}000,0\mathrm{.}000,0\mathrm{.}000]]$

Denote with $x_i$ the random variable that counts the number of goals that team iin $\{1,2\}$ will score.

a$.$ Calculate $]3,x_2>[2.$

b$.$ Calculate $F_{x_1,x_2}(2,1).$

c$.$ Find the marginal probability mass function of both $x_1$ and $x_2.$

d$.$ Are $x_1$ and $x_2$ stochastically independent? Why $($not$)?$

e$.$ What is the probability mass function of $x_1$ if it is known that team $2$ will score two goals?

f$.$ Find the probability mass function of $x_2$ conditioned on the event $\{x_1=x_1\}$ for every $x_{1}in\{0,1,2,3,4,5\}.$

g$.$ Apply results of part $f$ in the law of total probability to calculate $]=[0$ and $]=[3.$

h$.$ Use results of part $f$ in Bayes' formula to obtain $]=2|x_2|=[3.$

i$.$ Calculate the expectations of both $x_1$ and $x_2.$

j$.$ Calculate the variances of both $x_1$ and $x_2.$

k$.$ Calculate the covariance between $x_1$ and $x_2.$

l$.$ Calculate the correlation between $x_1$ and $x_2.$

m$.$ Calculate $|]=[3.$

n$.$ Calculate $[[x_1|]].$