Let X : Ω → R be a random variable on a probability space that is normally distributed.

Question

Let $X:\Omega\mapsto \Bbb R$ be a random variable on a probability space $(\Omega,\mathcal F,\Bbb P)$ that is distributed according tothe normal distribution with mean $\mu\in\Bbb R$ and variance $\sigma^2>0$. Let $M>0$ and show that $$\lim_{c\to\infty}\Bbb P(\{\lvert X\rvert>c\})e^{Mc}=0$$

This is the full question above.

I'm quite stuck on this question so I was just hoping for some hints on how to go about proving the limit at the bottom of the question.

Any help is greatly appreciated

Answer 1

if you first consider a standard normal variable, you could use $0 \le e^{Mc}\int_c^\infty e^{-x^2/2}dx \le e^{Mc} \int_c^\infty \frac{x}{c} e^{-x^2/2}dx$ which can be integrated easily using the substitution $u = x^2$.