# Derivative of entropy of a mixture of random variables

Setting

Consider two random variables $X_1, X_2$ and their entropy $H(X_1), H(X_2)$. Let $V$ be a random variable which takes values $1,2$ with equal probability.

Problem Statement

Let $\frac{dH(X_1)}{d\theta} = -1$ and $\frac{dH(X_2)}{d\theta} = -1$, can we say that $\frac{dH(X_V)}{d\theta} <= 0$ ?

• Do you mean $V$ takes values from $X_1$ and $X_2$ with equal prob? and what is $\theta$ ? – horaceT Sep 15 '16 at 22:56
• $X_V$ takes values from $X_1$ and $X_2$ with equal probability. P(X_v=x) = 0.5*P(X_1 = x )+ 0.5*P(X_2=x). $\theta$ is a parameter on which $X_1$ and $X_2$ depend. – Vivek Bagaria Sep 15 '16 at 23:22
• I could give more details about $\theta$ but..it will make the problem messy. – Vivek Bagaria Sep 15 '16 at 23:24