# Approximation of discrete distribution

I have a discrete random variable $X$ which takes the values $+1$ and $-1$ with equal probability $\frac{1}{2}$. Can I approximate this with a normal distribution ?

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Why would you want to do that, by the name of the Great Flying Spaghetti Monster? –  Did Jan 10 '13 at 17:05

It depends on the distance between probability distribution you are interested in. In your case the total variation distance will not provide any interesting results, because discrete distribution and continuous distribution are mutually singular, so the total variation will be always $1$.
However, you may have a weak convergence result by approximating two $\delta$-functions at $\pm 1$ with continuous densities. For example, you may take $X_n$ which is uniformly distributed on $$A_n = \left[-1-\frac1n,-1+\frac1n\right]\cup\left[1-\frac1n,1+\frac1n\right]$$ in the sense that its density if $f_n(x) = \frac n4\cdot1_{A_n}(x)$. However, I am not sure whether it helps you.