Using basic statistical properties of the variance, as well as single-variable calculus, derive (5.6). In other words, prove that α given by (5.6) does indeed minimize Var(αX + (1 − α)Y ).
The formula 5.6 is: \[ \alpha = \frac{\sigma^2_Y - \sigma_{XY}}{\sigma^2_X + \sigma^2_Y - 2\sigma_{XY}} \]
We will now derive the probability that a given observation is part of a bootstrap sample. Suppose that we obtain a bootstrap sample from a set of n observations.
What is the probability that the first bootstrap observation is not the jth observation from the original sample? Justify your answer.
There is a 1/n chance that the observation is in