standard deviation of a certain distribution

Question

If I have a list of N outcomes of drawing a number from the set {-1\$,+1\$}, and I know that the probability of getting (in a single draw) (-1\$) is p, and probability of getting (in a single draw) (+1\$) is (1-p), how can I use this information to calculate standard deviation for such a list? PS. The draws are independent from each other. Thanks in advance :-)

Answer 1

We interpret the question in a way that may not be the intended one. Let $X_1,X_2,\dots, X_N$ be independent random variables such that $\Pr(X_i=-1)=p$ and $\Pr(X_i=1)=1-p$. Let $Y=X_1+X_2+\cdots+X_N$. We want to find the standard deviation of the random variable $Y$.

The variance of a sum of independent random variables is the sum of the variances. So $$\operatorname{Var}(Y)=\operatorname{Var}(X_1)+\operatorname{Var}(X_2)+\cdots+\operatorname{Var}(X_N).$$

Now we compute the variances of the $X_i$. hey all have the same variance, so we find $\operatorname{Var}(X_1)$. This is $E(X_1^2)-(E(X_1))^2$.

Note that $X_1^2$ is identically equal to $1$, so $E(X_1^2)=1$. It follows that $\operatorname{Var}(X_1)=1-(1-2p)^2=4p(1-p)$.

It follows that $\operatorname{Var}(Y)=4Np(1-p)$, and therefore $Y$ has standard deviation $2\sqrt{Np(1-p)}$.

Answer 2

If you don't know $p$ then you could use the sample to estimate it, and calculate the corresponding standard deviation. But if you know $p$, then there's no uncertainty in the distribution itself, so there's no standard deviation to calculate there. The title implies that you do know $p$, so what standard deviation are you talking about?