Thread: How to find the mean and variance of new combinations of old population?

I'm a late learner of maths &science.
Any books discuss these kind of problems?

3 groups of data: $\displaystyle f_ix_i,\ g_iy_i,\ h_iz_1$

f g h are their frequencies.
mean: $\displaystyle \bar{x},\ \bar{y},\ \bar{z}$
variance: $\displaystyle \sigma_x,\ \sigma_y,\ \sigma_z$

Find the mean and variance of $\displaystyle T(f_ix_i,\ g_iy_i,\ h_iz_i)$

T can be many different operations.

for example:

T may be: $\displaystyle ax_i+b$

T may be: $\displaystyle ax_i+by_i$

T may be: $\displaystyle e^{x_i}$

T may be: $\displaystyle lnx_i$

T may be: $\displaystyle x_i+x_j$, mean of two elements drawn from $\displaystyle f_ix_i$ with replacement i=j is allowed. $\displaystyle n^2$

T may be: $\displaystyle x_i+x_j$, two elements drawn from $\displaystyle f_ix_i$ without replacement where $\displaystyle i\ \neq\ j$. $\displaystyle C^n_2$

T may be: [????I don't know how it looks like.] is $\displaystyle P^n_2$

T may be: $\displaystyle x_i+x_j+x_k$, three elements drawn from $\displaystyle f_ix_i$ without replacement where $\displaystyle i\ \neq\ j\ \neq\ k$. $\displaystyle C^n_3{$

T may be: $\displaystyle ax_i+by_i+cz_i$, a b c elements drawn from $\displaystyle f_ix_i,\ g_iy_i,\ h_iz_1$ respectively with replacement. $\displaystyle n^{a+b+c}$

T may be: [????I don't know how it looks like.], a b c elements drawn from $\displaystyle f_ix_i,\ g_iy_i,\ h_iz_1$ respectively without replacement. $\displaystyle [????$I can't figure out the combination]