Suppose that 200 points are selected independently and at random from the unit s

Question

Suppose that 200 points are selected independently and at random from the unit square:S={(x,y):0<=x<1,0<=y<1}. LetA={(x,y): x+y<=0.5}be a triangle representing a sub area of S.

(a) What is the probability that a randomly selected point falls into A?

(b) Suppose n points are randomly selected from S, and let X be the number of them fall into area A, what is the distribution of X? (b) How many points out of 200 you expect to fall into the area A?

(c) What are the variance and standard deviation of X, given n = 200?

(d) What is probability that out of 10 points selected, 4 or 5 of them fall into A?

Explanation / Answer

(a) Here

Pr(a randomly selected point falls into A) = Area of A/ Area of S = (1/2 * 0.5 * 0.5)/ (1 * 1) = 0.125

(b) Here the distribution of X would be binomial distribution with expected mean = n * 0.125

Expected to fall into area A = 200 * 0.125 = 25

(c) Variance of X = 200 * 0.125 * 0.875 = 21.875

standard deviation = sqrt(21.875) = 4.677

(d) Here n = 10

Pr( X = 4 or 5 ; 10 ; 0.125) = Pr(X = 4 ; 10 ; 0.125) + Pr(X = 5 ; 10 ; 0.125)

= 10C4 (0.125)4(0.875)6 + 10C5 (0.125)5 (0.875)5

= 0.0230 + 0.0039

= 0.0269

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