When writing down the procedure hypothesis tests, I try to follow the convention that upper case letters $X,Y$ represents random variables (RV) and small letters $x,y$ represents the outcome of ONE experiment, i.e., a sample. However, I find that I am having some difficulties doing this due to the lack of notation.
To give an example from two sample $t$-test:
From the data given, the unbaised estimators $\hat\sigma^2_1=56.4$, $\hat\sigma^2_2=53.4$. (Just an example, numbers are randomly given.) Therefore, the pooled estimate of the variance of samples (assuming the both samples have the same variance) is $$s_p^2=\frac{(n_1-1)\hat\sigma^2_1+(n_2-1)\hat\sigma^2_2}{n_1+n_2-2}=54.0 $$ The test statistic is $$t=\frac{\bar y-\bar x}{s_p(1/n_1+1/n_2)^{1/2}}=1.24 $$ Now, $(n_k-1)\hat\sigma^2_k/\sigma^2$ follows a $\chi^2$ distribution of freedom degree $n_k-1$, ... (to be continued)
Here I run into a problem. I already give $\sigma^2_1$ a specific value $56.4$, so I cannot really say that $(n_k-1)\hat\sigma^2_k/\sigma^2$ follows a $\chi^2$ distribution. I must replace $\sigma^2_k$ by an upper case letter, $\Sigma^2_k$, to distinguish between the value obtained from one experiment and the RV itself. However, I never see $\Sigma^2_k$ before - it looks very strange.
Let's continue.
Let $\sigma$ be the common variance of $X$ and $Y$. Now, $(n_k-1)\hat\sigma^2_k/\sigma^2$ follows a $\chi^2$ distribution of freedom degree $n_k-1$, so $(n_1+n_2-2 )S_p^2 /\sigma^2$ follows a $\chi^2$ distribution of freedom degree $n_1+n_2-2$. Therefore we may conclude that $$ T=\frac{\bar Y-\bar X}{S_p(1/n_1+1/n_2)^{1/2}} $$ follows a student distribution of freedom degree $\nu=n_1+n_2-2$. We can now check the critical value from the table...
The rest should be no problem. For $X,Y,S_p$, capital letters are quite easy to use.
Question: What should I use for the "upper case" $\hat \sigma$? If I use $S$, then it might be confused with the moments.