Log Normal Mean's Logarithm

This quant interview question is about understanding the lognormal distribution that arises when you exponentiate a normal random variable. It checks whether you can move comfortably between the distribution of a Gaussian variable, its exponential, and the expectations of transformed variables. In quant prep, this sits at the intersection of continuous random variables, measure-theoretic expectations, and basic stochastic calculus intuition.

It trains your grasp of how nonlinear transformations affect expectations, especially when going from normal to lognormal. You must know how variance and mean both feed into expectations under exponentials, and how to use the right distributional properties instead of naïve intuition. It deepens your fluency with continuous distributions and their moments.

This matters for quant interviews because pricing, risk, and volatility modeling often assume lognormal dynamics. Being precise about expectations under lognormality is core to option pricing, PnL modeling, and many practical quant finance calculations.

What it tests

Whenever a random variable is defined as the exponential of a normal variable, such as $X = e^Z$ with $Z \sim N(\mu, \sigma^2)$, the expectation $\mathbb{E}[X]$ is not simply $e^{\mathbb{E}[Z]}$ due to the nonlinearity of the exponential function. Instead, it is governed by the moment-generating function (MGF) of the normal distribution: $\mathbb{E}[e^{\theta Z}] = e^{\mu \theta + \frac{1}{2}\sigma^2 \theta^2}$. This arises because the exponential function amplifies the effect of variance, not just the mean, so the expectation incorporates both. The key is that the MGF encodes all moments of the distribution, and for the normal, it has a simple closed form. This principle generalizes: for any function $g(Z)$, $\mathbb{E}[g(Z)]$ depends on the distribution of $Z$ and the form of $g$, and for exponentials, the MGF is the tool that captures the effect of both mean and variance.