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

1. 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: $f_ix_i,\ g_iy_i,\ h_iz_1$

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

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

T can be many different operations.

for example:

T may be: $ax_i+b$

T may be: $ax_i+by_i$

T may be: $e^{x_i}$

T may be: $lnx_i$

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

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

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

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

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

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