uppose X $$ tp$4$q$,$ Y $$ tp$200$q$,$ and Z $$ Np$0,1$q$.$ Answer the following questions. Attach R code if used.

$($a$)$ Compute PpX $2$q$,$ PpY $2$q$,$ and PpZ $2$q$.$

$($b$)$ Compare and contrast the probabilities you got in part $($a$).$ Which one has largest probability? Which one has the smallest probability? Which ones are close to each other? Which ones are different from each other?

$($c$)$ Compute Pp $0\mathrm{.}5$ X $0\mathrm{.}5$q$,$ Pp $0\mathrm{.}5$ Y $0\mathrm{.}5$q$,$ and Pp $0\mathrm{.}5$ Z $0\mathrm{.}5$q$.$

$($d$)$ Compare and contrast the probabilities you got in part $($c$).$ Which one has largest probability? Which one has the smallest probability? Which ones are close to each other? Which ones are different from each other?

$($e$)$ Based on your answers in the previous parts, complete the following sentence. $$Compared with the standard normal distribution and the t$-$distribution with a large degrees of freedom, the t$-$distribution with a small degrees of freedom has

tails and center.$$