excans:B4a44e0a2b

Answer: D

This is a mixed distribution for the population, since the vaccine will apply to all once available.

Available?

[math](A)[/math]	[math]\operatorname{Pr}(A)[/math]	[math]{ }_{2} p \mid A[/math]	[math]E(S \mid A)[/math]	[math]\operatorname{Var}(S \mid A)[/math]	[math]E\left(S^{2} \mid A\right)[/math]
Yes	0.2	0.9702	97,020	2,891	[math]9,412,883,291[/math]
No	0.8	0.9604	96,040	3,803	[math]9,223,685,403[/math]
			[math]E(S)[/math]		[math]E\left(S^{2}\right)[/math]
			96,236		[math]9,261,524,981[/math]
		[math]\operatorname{Var}(S)[/math]	157,285
		[math]S D(S)[/math]	397

As an example, the formulas for the "No" row are

[math]\operatorname{Pr}(\mathrm{No})=1-0.2=0.8[/math]

[math]{ }_{2} p[/math] given [math]\mathrm{No}=(0.98[/math] during year 1[math])(0.98[/math] during year 2[math])=0.9604[/math]

[math]E(S \mid \mathrm{No}), \operatorname{Var}(S \mid[/math] No [math])[/math] and [math]E\left(S^{2} \mid\right.[/math] No [math])[/math] are just binomial, [math]n=100,000 ; \mathrm{p}([/math] success [math])=0.9604[/math]

[math]E(S), E\left(S^{2}\right)[/math] are weighted averages,

[math]\operatorname{Var}(S)=E\left(S^{2}\right)-E(S)^{2}[/math]

Or, by the conditional variance formula:

[[math]] \begin{aligned} \operatorname{Var}(S) & =\operatorname{Var}[E(S \mid A)]+E[\operatorname{Var}(S \mid A)] \\ & =0.2(0.8)(97,020-96,040)^{2}+0.2(2,891)+0.8(3,803) \\ & =153,664+3,621=157,285 \\ \operatorname{StdDev}(S) & =397 \end{aligned} [[/math]]

Copyright 2024 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.