$1.$ Joe faces the following loss distribution for his annual spending on personal medical expenses:

LOSS $0100020003000$

PROBABILITY $\mathrm{.}20\mathrm{.}30\mathrm{.}30\mathrm{.}20$

$1$a$.$ Calculate the mean and standard deviation of Joe's loss distribution.

$1$b$.$ Assume that Joe agrees to pool his medical expense losses with those of Lee, who lives $3000$ miles away from Joe. Lee$$s loss distribution has the following characteristics:

Lee$$s mean loss $=$ $$3000$

Lee$$s standard deviation $=$ $$1025$

Describe the assumptions underlying the risk pooling model $($homogeneity and independence$),$ and explain if the assumptions are satisfied in this pooling arrangement

$1$c$.$ Consider Insurer XXX$,$ which has insured $2500$ independent and homogeneous people identically distributed to Lee. Calculate the expected value of the insurer$$s mean loss per policy, and the corresponding standard deviation of the mean loss distribution.

$1$d$.$ Calculate a $96\%$ confidence interval for Lee's estimate of the mean loss, assuming that the underlying distribution from which the above losses are drawn is unknown.