Consider $$ i$.$i$.$d$.$ random variables

$,=1,...$

each with mean $$ and variance $2$

$.$

a$.$ Show directly that

$()=2$

$$

$,$

where$$ is the sample mean.

b$.$ Now consider $$ identically distributed random variables $,=1,...$ each with mean

$,$ variance $2$ and pairwise positive correlation $,$ show that

$()=2+1$

$2$

$.$

$$

You may assume that

$($

$$

$=1$

$)2$

$=$

$$

$$

$=1$

$$

$$

$=1$

$$

c$.$ Explain why the result from Part $($b$)$ causes a problem with bagged trees, and explain

how the random forest algorithm overcomes this problem.