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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
up vote 1 down vote accepted

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.

As Eckhard told you, with the single normal distribution you never achieve a good quality of the approximation since it's either have a different mean, or its variance is too big to approximate the discrete distribution. Maybe if you provide more details, what is the motivation for your question, there will be more to say.

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Of course, but the approximation will not be very good.

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