Question

3 Probability and Statistics [10 pts] Consider a sample of data S obtained by flipping a coin five times. X,,i e..,5) is a random variable that takes a value 0 when the outcome of coin flip i turned up heads, and 1 when it turned up tails. Assume that the outcome of each of the flips does not depend on the outcomes of any of the other flips. The sample obtained S - (Xi, X2,X3, X, Xs) (1, 1,0,1,0 (a) What is the sample mean for this data? (b) What is the unbiased sample variance? (c) What is the probability of observing this data assuming that a coin with an equal probability of heads and tails was used? (i.e., The probability distribution of Xi is P(X-1) -0.5 P(X0) 0.5.) d) Note the probability of this data sample would be greater if the value of the probability of heads P(X1 was not 0.5 but some other value. What is the value that maximizes the probability of the sample S? Optional: Can you prove your answer is correct? e) Given the following joint distribution between X and Y, what is P(x TY- b)? 0.2 0.2 F0.05 0.15 0.3

Answer 1

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