The probability density function of a continuous random variable \(X\) that ranges from 1 to \(e^{2}\) is given by
\[f(x)=\frac{1}{2x}.\]
Find the value of \(k\) such that \(P(X>k)=\frac{1}{10}.\)

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If the probability density function of a continuous random variable \(X\) is given by
\[f(x)=\begin{cases} ax^2\ &(0\le x\le1) \\-\frac{a}{9}(x-1)+a\ &(1<x\le10),\end{cases}\]
what is the value of \(a?\)

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If the probability density function of a continuous random variable \(X\in[0,\frac{\pi}{16}]\) is given by
\[f(x)=a\sin8x,\]
what is the variance of \(X?\)

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