Question

The differential entropy of a Gaussian random variable with zero mean and variance of 2 will be

• A. $1+\frac{1}{2}lo{g}_2\left(\pi e\right)$
• B. $\frac{1}{2}+\frac{1}{2}lo{g}_2\left(\pi e\right)$
• C. $\frac{1}{2}lo{g}_2\left(\pi e\right)$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $1+\frac{1}{2}lo{g}_2\left(\pi e\right)$

For zero mean Gaussian random variable, the differential entropy can be given by,
H(X) = $\frac{1}{2}lo{g}_2\left(2\pi e{\sigma }^2\right)$

Given that, ${\sigma }^2=2$

So, H(X) = $\frac{1}{1}lo{g}_2\left(4\pi e\right)=\frac{1}{2}lo{g}_2\left(\pi e\right)$

$=1+\frac{1}{2}lo{g}_2\left(\pi e\right)$