Q$2)(25\%)$ Due to climate change, temperatures and flood risk projections for the end of the century $($i$.$e$.,2090-2100)$ indicate higher values compared to present conditions. Given the significant uncertainty surrounding these projections, the rates of increase in temperature and flood risk are estimated probabilistically. Consequently, there exists a well$-$established correlation between rate of increase in temperature $($T$,$ measured in $)$ and flood risk $($F$,$ represented as a percentage$).$ The joint probability mass function $($PMF$)$ depicting the relationship between T and F across a specified region for the end of the century is presented in the following table. Specifically, the projected temperature increase range spans from $1$ to $5,$ while the flood risk increase range extends from $10\%$ to $50\%.$

T$/$F $1020304050$

$10\mathrm{.}120\mathrm{.}020\mathrm{.}000\mathrm{.}000\mathrm{.}00$

$20\mathrm{.}020\mathrm{.}180\mathrm{.}030\mathrm{.}010\mathrm{.}00$

$30\mathrm{.}000\mathrm{.}030\mathrm{.}240\mathrm{.}000\mathrm{.}00$

$40\mathrm{.}000\mathrm{.}010\mathrm{.}000\mathrm{.}180\mathrm{.}02$

$50\mathrm{.}000\mathrm{.}000\mathrm{.}000\mathrm{.}020\mathrm{.}12$

$(4\%)$ a$)$ Determine the marginal distributions of T$.$

$(4\%)$ b$)$ Determine the marginal distributions of F$.$

$(5\%)$ c$)$ Find the probability that the temperatures will increase $2$ or more and flood risk will increase $30\%$ or more at the end of century compared to today$$s values.

$(4\%)$ d$)$ Find the probability that the temperatures will increase $4$ or more given flood risk will increase $30\%$ or more at the end of century compared to today$$s values.

$(4\%)$ e$)$ Calculate the expected values of T and F$.$

$(4\%)$ f$)$ Are T and F statistically independent? Show.