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An insurance company has $10,000$automobile policyholders. The expected yearly claim per policyholder is$240$, with a standard deviation of$800$.

Let $X_i$represents the yearly claim perish policyholder,$i=1,2,$\dots,$10,000$It is given that and${蟽}^2=Var\left(X_i\right)={800}^2=640,000$.

Let's $脳$denote the total yearly claim$X=\underset{1}{\overset{10,000}{鈭�}}{X}_i$.

Because of the independence of random variables${\mathrm{X}}_{-}$, using the corresponding properties of expectation and variance we get$E\left[X\right]=(10,000)渭=2,400,000$, $Var\left(X\right)=(10,000){蟽}^2=6,400,000,000$

The probability that the total yearly claim exceeds $2.7$million dollars is

$P\{X>2,700,000\}.$To approximate this probability we use the central limit theorem and in that case, we get$$\begin{gathered} :P\{X>2,700,000\}==1-P\{X鈮�2,700,000\}=1-P\left\{\frac{X-E\left[X\right]}{\sqrt{Var\left(X\right)}}鈮�\frac{2,700,000-E\left[X\right]}{\sqrt{Var\left(X\right)}}\right\} \\ =1-P\left\{\frac{X-2,400,000}{\sqrt{6,400,000,000}}鈮�\frac{2,700,000-2,400,000}{\sqrt{6,400,000,000}}\right\}鈮�1-桅(3.75) \\ \text{software package}1-.9999116=\mathbf{0}\mathbf{.}\mathbf{0000884}\mathbf{鈮�}\mathbf{0} \end{gathered}$$