# Statistics

## MSE of Sample Variance

In introductory statistics courses, we often focus on unbiased estimators such as the sample variance, $$S^2 = \frac{\sum(X_i - \bar X)^2}{n - 1}$$ rather than a more initiative estimator  \hat \sigma^2 = \frac{\sum(X_i - \bar X)^2}{n}.

## Variance of sample mean squared

It is well known that $\bar X^2$ is a biased estimator for $\mu^2$, but a little is known to its variance. Before we start working on $Var(\bar X^2)$, let’s recall that $\bar X$ is an unbiased estimator for $\mu$, i.

## Girl ratio

https://stats.stackexchange.com/questions/93830/expected-number-of-ratio-of-girls-vs-boys-birth The Zorganian Republic has some very strange customs. Couples only wish to have female children as only females can inherit the family’s wealth, so if they have a male child they keep having more children until they have a girl.

## Two cool things about dice!

Just discover two cool things about dice, Suppose we have two (fair) 6-sided dice, let $Y$ be the reminder of the product of the two numbers divided by 7, i.e., $Y = X1 \times X2 (\text{ mod } 7)$, guess what the distribution of $Y$ is?