Let $X$ be a standard normal random variable. Then, for any differentiable f:R→R such that $\mathbb{E}f(X)^2<∞$ the Gaussian Poincare inequality states that$$Var(f(X))≤\mathbb{E}[f′(X)^2]$$. I'd like to know what is the equivalent bound for normal random variables with variance $\sigma^2 \neq 1$, if there is such an equivalent. I tried reading up on the Gaussian Poincare inequality and where the fact that $\sigma^2 = 1$ comes in, and couldn't find my way around it. I'd love some help. Thanks!
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