Variance of log X for a Lognormal Distribution | GATE ST 2025 | Problem - 40 | RitwikMath

Описание: Given a continuous random variable \(X\) with lognormal distribution parameters \(\mu \in \mathbb{R}\) and \(\sigma 0\), and the condition:
\[
\ln \left(\frac{E[X^2]}{(E[X])^2}\right) = 4,
\]
find \(\mathrm{Var}(\ln X) = \sigma^2\).

Key steps:
If \(Y = \ln X \sim N(\mu, \sigma^2)\), then
\[
E[X] = E[e^Y] = e^{\mu + \frac{\sigma^2}{2}},
\]
and
\[
E[X^2] = E[e^{2Y}] = e^{2\mu + 2\sigma^2}.
\]
Calculate the ratio inside the log:
\[
\frac{E[X^2]}{(E[X])^2} = \frac{e^{2\mu + 2\sigma^2}}{(e^{\mu + \frac{\sigma^2}{2}})^2} = e^{\sigma^2}.
\]
Taking the natural logarithm:
\[
\ln \left(\frac{E[X^2]}{(E[X])^2}\right) = \sigma^2.
\]
Given this equals 4, so
\[
\sigma^2 = 4.
\]

Therefore, the variance of \(\ln X\) is 4.

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