Prove that a Truncated Gaussian is log-concave

Question

Is the truncated Gaussian a log-concave distribution?

I read here at Section 2.2.5 that whatever log-concave truncated distribution (and hence also Gaussian) is log-concave and the proof is trivial. How to prove this statement?

Answer 1

the density of a truncated rv is the following

$$f_{X|X>c}=\frac{1}{1-F_X(c)}f_X(x)$$

This prove your statement

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Proof of the truncated density

Let's$X$ a rv (assuming it is continuous) with CDF $F_X(x)$ and let $B=\{X>c\}$

Thus, by definition

$$F(x|B)=\frac{\mathbb{P}[c<X<x]}{\mathbb{P}[X>c]}=\frac{F_X(x)-F_X(c)}{1-F_X(c)}$$

if $x>c$ and zero if $x\leq c$

derivating F you get the density I showed before