SDS321HW22Sol

# SDS321HW22Sol - SDS321HW22Sol 1 The probability of...

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SDS321HW22Sol 1. The probability of obtaining heads in a single flip of a certain coin is a random variable, denoted by Θ , which is uniformly distributed in [0, 1]. Let X be a Bernoulli random variable with X= 1 if the coin flip results in heads, and X = 0 if the coin flip results in tails. (First three are from a SDS321 Final) a. Find the mean and variance of X. E X [ ] = E E X | Θ [ ] " # \$ % = E Θ [ ] = 1 2 Var X [ ] = Var E X | Θ [ ] " # \$ % + E Var X | Θ [ ] " # \$ % = Var ( Θ ) + E [ Θ (1 −Θ )] = 1 12 + ( 1 2 ( 1 12 + ( 1 2 ) 2 ) = .25 b. Find the covariance of X and Θ . E X Θ [ ] = ( X Θ f ( x , Θ ) d Θ 0 1 X = 0 1 ) = 0 + ((1* Θ )* Θ ) d Θ 0 1 = 1 3 E X [ ] = E E X | Θ [ ] \$ % & ' = E Θ [ ] = 1 2 E Θ [ ] = 1 2 Cov ( X , Θ ) = E X Θ [ ] E X [ ] E Θ [ ] = 1 3 1 2 × 1 2 * + , - . / = 1 12 c. Find the conditional PDF of Q given that X = 1. f Θ | X = 1 θ ( ) = f Θ , X θ ,1 ( ) p X 1 ( ) = p X | Θ 1 ( ) f Θ ( θ ) p X 1 ( ) = 2 θ 0 θ 1 Which you may notice is a special case of a Beta distribution. Beta(2,1) d. Determine the posterior distribution of Θ given in 10 trials, there were 10 tosses of heads if a uniform (0,1) prior is used. L ( Θ ) = P ( data | Θ ) ∝Θ 10 0 Θ≤ 1 p ( Θ ) 1 0 Θ≤ 1 p ( Θ | data ) ∝Θ 10 0 Θ≤ 1

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SDS321HW22Sol This is a beta(11,1). e. Give the MAP estimate for Θ . This is a beta(11,1).
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