Problem 2. Answer the following: (a) Let W be a random variable giving the numbe

Question

Problem 2. Answer the following: (a) Let W be a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of the sample space for the three tosses and to each sample point assign a value w of W. (b) Find the p.m.f. and c.d.f. of W, assuming that the coin is biased so that a head is twice as likely to occur as a tail. (c) Plot the probability histogram of W. You should use rectangles of width 2 centered at each value x (so there will be no gaps between your rectangles). You should also adjust the height of each rectangle so that the area of the rectangle associated with the value w, for example, is equal to P(W = w). This means the areas of the rectangles should sum to 1.

Explanation / Answer

a)

W = number of head - number of tails in 3 tosses of a coin

w can be -3 , -1 , 1 , 3

b) P(x,y) = Probability of x head and y tails

X = number of heads

this is binomial distribution with   n = 3 , p = 2/3

P(H) = 2/3 , P(T) = 1/3

P(W = -3) = P(X =0) = (1/3)^3 = 1/27

P(W = -1) = P(X =1) = 3 * 2/3 * (1/3)^2 = 6/27

P(W =1) = P(X =2) = 3 * (2/3)^2 * 1/3 = 12/27

P(W = 3) = P(X = 3) = (2/3)^3 = 8/27

cdf of W =

1/27    W <= -3

7/27   W <= -1

19/27 W<= 1

1   W

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