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We're dealing with the lognormal distribution in a finance course and my textbook just states that this is true, which I find sort of frustrating as my maths background isn't very strong but I want the intuition. Can anyone show me why this is the case?

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Recall that $e^x\geq 1+x$

$E\left[e^{Y}\right]=e^{ E(Y)} E\left[e^{Y- E(Y)}\right]\geq e^{E(Y)} E\left[1+{Y- E(Y)}\right] = e^{E(Y)}$

So $e^{E(Y)}\leq E\left[e^{Y}\right] $

Now letting $Y=\ln X$, we have:

$e^{E(\ln X)}\leq E\left[e^{\ln X}\right]=E(X)$

now take logs of both sides

$E[\ln (X)]\leq\ln[E(X)]$


Alternatively:

$\ln X = \ln X - \ln \mu+\ln\mu \qquad$ (where $\mu=E(X)$)

$\qquad= \ln(X/\mu)+\ln \mu $

$\qquad= \ln[ \frac{X-\mu}{\mu} + 1]+\ln \mu$

$\qquad \leq \frac{X-\mu}{\mu} + \ln \mu\qquad$ (since $\ln(t+1)\leq t$)

Now take expectations of both sides:

$E[\ln(X)] \leq \ln\mu$


An illustration (showing the connection to Jensen's inequality):

(Here the roles of X and Y are interchanged so that they match the plot axes; better planning would have swapped their roles above so the plot more directly matched the algebra.)

scatter plot of y=exp(x) vs x for a sample, showing the inequality arising from the curvature in that relationship

The solid coloured lines represent means on each axis.

As we see because the relationship "bends toward" $X$ in the middle (and "away from" $Y$), the mean of $Y$ (orange horizontal line) goes along a little further before hitting the curve (giving the small gap (marked in blue) between log(mean(y)) and mean(log(y)) that we see).

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