See Exercise 4.22. Using indicators, find the variance of the number of black two-by-two sub…

See Exercise 4.22. Using indicators, find the variance of the number of black two-by-two sub boards. Note that for the natural choice of indicators, the random variables are not independent.

Exercise 4.22

Take an n-by-n board divided into one-by-one squares and color each square white or black uniformly at random. That is, for each square flip a coin and color it white, if heads, and black, if tails. Let X be the number of two-by two square “sub boards” of the chessboard that are all black (See Fig. 4.4). Use indicator variables to find the expected number of black two-by-two sub boards.

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