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Does $\Bbb{E}(X^2)$ DNE $\Rightarrow \operatorname{Var}(X)$ DNE?

Suppose you have pdf $$f(x) = \begin{cases} \frac{8}{x^3} &, \text{ if $x\ge 2$} \\ 0 &, \text{ otherwise} \end{cases}$$ I have found that $\Bbb{E}(X)=4$ and am trying to find $\operatorname{Var}(X)$ using $\Bbb{E}(X^2)-(\Bbb{E}(X))^2$. To find $\Bbb{E}(X^2)$, I’ve been using $$\int_{-\infty}^\infty u^2 f(u) du = \int_2^\infty \frac{8}{u} du = \lim_{t\to \infty}(8\ln t – 8\ln2)$$ However, $\lim_{t\to \infty}(\ln […]

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Does $\Bbb{E}(X^2)$ DNE $\Rightarrow$ Var$(X)$ DNE?

Suppose you have pdf $f(x) = \begin{cases} \frac{8}{x^3}, & \text{if $x\ge 2$} \\ 0, & \text{otherwise} \end{cases}$. I have found that $\Bbb{E}(X)=4$ and am trying to find Var$(X)$ using $\Bbb{E}(X^2)-\Bbb{E}(X)$. To find $\Bbb{E}(X^2)$, I’ve been using $$\int_{-\infty}^\infty u^2 f(u) du = \int_2^\infty \frac{8}{u} du = \lim_{t\to \infty}(8\ln t – 8\ln2)$$ However, $\lim_{t\to \infty}(\ln t)$ DNE, […]