Smallest possible MGF value

HardRandom Variables Statistics~9m

A random variable $X$ has moment generating function $M(\theta) = \mathbb{E}[e^{\theta X}]$ , and $M(1) = 7$ . Over all such $X$ , what is the smallest possible value of $M(4)$ ?

Show hints (2)+

$\log M(\theta)$ is convex, and $\log M(0)=0$ , $\log M(1)=\log 7$ .
Extrapolate the chord to $\theta=4$ : $\log M(4)\ge 4\log 7$ , i.e. $M(4)\ge 7^4$ (a constant $X$ achieves it).

Answer

Reveal answer →

2401

Want the full step-by-step worked solution? It's part of Premium - along with a worked solution for every question in the bank.

Practise free Unlock solutions

Asked at: Jane Street, Two Sigma

Smallest possible MGF value

Answer

Related questions