Question: In random forests,Bbootstrap samples are obtained from the training data. A tree is built for each bootstrap sample. At each split of a tree, a

In random forests,Bbootstrap samples are obtained from the training data. A tree is built for each bootstrap sample. At each split of a tree, a random selection ofm predictors is chosen as candidates for the split. Suppose there areppredictors in the training data. The probability that a specific predictor,Xj, is not among them selected in a split is(p-m)/p.

(a) What is the probability thatXjis not the first selection, among thempredictors selected?

(b)Given thatXjwas not the first selection among themselected, what is the probability thatXjis not the second selection?

(c)Given thatXjwas neither the first nor the second selection among themselected, what is the probability thatXJis not the third selection?

(d)Use the previous parts to help you show that the probability thatXjis not among themselected predictors in a split isp.

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