# two definition of variance identity

$$I_n$$ is the indicator rv where $$I_n=1$$ if $$A_n$$ occurs and 0 otherwise.

Then define $$\eta_n = \sum_{k=1}^n I_k$$

I'm given this statement:

But in another post about the same proof, I have this:

$$X_n$$ is the same r.v. as $$\eta_n$$

But in the first case, $$\sigma^2(\eta_n)$$ is the standard deviation but in the second case, $$V(X_n)$$ is variance.

Are variance and standard deviations supposed to be equal?

This is the link to the first case page 8

This is the link to the second case enter link description here

## 1 Answer

Usually, the standard deviation is denoted with $$\sigma$$ and it is the square root of the variance. It seems to me that there is a mistake in the upper statement: the standard deviation should be $$\sigma$$ (without the square); so they should either have written "we denote the standard deviation by $$\sigma(X)$$" or "we denote the variance by $$\sigma^2(X)$$". That is the only way the equation is correct anyway.

• thanks for the clarification! – user8290579 Mar 3 at 23:45