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GO Classes DPP | STATISTICS |VARIANCE

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Suppose $X_i \sim \operatorname{Normal}\left(0, \frac{1}{i^2}\right)$, where $i=1,2, \ldots, 9$ and $X_1, X_2, \ldots, X_9$ are independent to each other. Let $Y$ be a random variable defined as $Y=\sum_{i=1}^9 i X_i$. Find the variance of $Y$.

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Since Xi 's are independent, hence $\operatorname{var}(\mathrm{Xi}+\mathrm{Xj})=\operatorname{var}(\mathrm{Xi})+\operatorname{var}(\mathrm{Xj}) \operatorname{bcz} \operatorname{cov}(\mathrm{Xi}, \mathrm{Xj})=0$

$$
\begin{aligned}
\operatorname{Var}(Y) & =\operatorname{Var}\left(\sum_{i=1}^9 i X_i\right) \\
& =\operatorname{Var}\left(X_1+2 X_2+3 X_3+\ldots+9 X_9\right) \\
& =\operatorname{Var}\left(X_1\right)+\operatorname{Var}\left(2 X_2\right)+\ldots+\operatorname{Var}\left(9 X_9\right) \\
& =\operatorname{Var}\left(X_1\right)+4 \operatorname{Var}\left(X_2\right)+\ldots+81 \operatorname{Var}\left(X_9\right) \\
& =\frac{1}{1^2}+4\left(\frac{1}{2^2}\right)+\ldots+81\left(\frac{1}{9^2}\right) \\
& =9
\end{aligned}
$$
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