91影视

a fair die is rolled $100$times. $X_i$be the value obtained on the,$i$th roll.

Assume that a fair die is rolled$100$times and let$X_i$represent the value obtained on the $i$th roll. Then, the random variables $X_1,X_2,鈥�,X_{100}$are independent identically distributed, with mean

$渭={渭}_i=E\left(X_i\right)=\frac{1}{6}(1+2+3+4+5+6)=\frac{7}{2}$

and variance

${蟽}^2={蟽}_i^2=Var\left(X_i\right)=\frac{1}{6}\left(1^2+2^2+3^2+4^2+5^2+6^2\right)-{渭}^2=\frac{35}{12}$

Therefore, the random variables $\ln \left(X_1\right),\ln \left(X_2\right),鈥�,\ln \left(X_{100}\right)$are independent identically distributed, with mean

${渭}_l=E\left(\ln \left(X_i\right)\right)=\frac{1}{6}\left(\ln \right(1)+\ln (2)+\ln (3)+\ln (4)+\ln (5)+\ln (6\left)\right)鈮�1.1$

and variance

localid="1649860717881" ${蟽}_l^2=Var\left(\ln \left(X_i\right)\right)$$=\frac{1}{6}\left(\ln (1{)}^2+\ln (2{)}^2+\ln (3{)}^2+\ln (4{)}^2+\ln (5{)}^2+\ln (6{)}^2\right)$$-{渭}_l^2鈮�0.37$

Further, let$1<a<6$. We have:

$$\begin{gathered} P\left\{\underset{1}{\overset{100}{鈭�}}{X}_i鈮�a^{100}\right\}=P\left\{\ln \left(\underset{1}{\overset{100}{鈭�}}{X}_i\right)鈮�\ln \left(a^{100}\right)\right\}= \\ P\left\{\underset{1}{\overset{100}{鈭�}}\ln \left(X_i\right)鈮�100\ln \left(a\right)\right\}= \\ P\left\{\frac{{鈭�}_1^{100}\ln \left(X_i\right)-100{渭}_l}{{蟽}_l\sqrt{100}}鈮�\frac{100\ln \left(a\right)-100{渭}_l}{{蟽}_l\sqrt{100}}\right\} \end{gathered}$$

The central limit theorem

$桅\left(\frac{100\ln \left(a\right)-100{渭}_l}{10{蟽}_l}\right)$