hw1_sol

# 0 0 000 2 4 004 p y57 6 006 8 008 002 pp

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Unformatted text preview: nsity of θ, i.e., the un-normalized function. The plot in (e), on the other hand, corresponds to the normalized posterior density function. These plots suggest that the most likely value of θ is 0.57. 3 0.7 0.6 0.06 0.2 0.3 p(θ|∑ y=57) 0.4 0.5 0.05 0.04 0.03 p(∑ y=57|θ) 0.02 0.1 0.01 0.0 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 1.0 0.8 1.0 0 0.00 2 4 0.04 p(θ|∑ y=57) 6 0.06 8 0.08 θ 0.02 p(θ)*p(∑ y=57|θ) 0.8 0.6 θ 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 θ θ ￿ Figure 1: The plots correspond to: p( i Yi = y |θ) versus θ for (b); p(θ| ￿ ￿ p(θ) × p( n Yi = 57|θ) versus θ for (d); p(θ| n Yi = 57) for (e) i=1 i=1 ￿ i Yi = 57) versus θ for (c); 3. Suppose the number of defects in a 1200-foot roll of magnetic recording tape has a Poisson distribution with mean θ. Let the prior distribution of θ be Gamma(3, 1). When ﬁve rolls of this tape are selected at random and inspected, the number of defects found on the rolls are 2, 2, 6, 0, and 3. (a) Determine the posterior distribution of θ. The likelihood function is given by L( θ ) = n ￿ e − θ θ yi yi ! i=1 = e −n θ θ ￿n i=1 yi i=1 yi ! and the prior distribution of θ is π (θ ) = ￿n ba a − 1 − b θ θe Γ(a) Thus, the posterior distribution of θ is p( θ | y ) ∝ e − n θ θ 4 ￿n i=1 yi a − 1 − b θ θ e =θ ￿n i=1 yi + a − 1 − θ (n + b ) e We recognize the kernel of a Gamma density θ|y ∼ Gamma(a + n ￿ yi , b + n) i=1 Note that the posterior mean of θ can be presented as the weighted average of the MLE and the pior mean: ￿ ￿n a + n yi n ba i=1 i=1 yi E [θ |y ] = = + b+n b+n n b+nb For the observed data, n = 5 and thus, ￿5 i=1 yi = 13, and the hyperparameters were set to a = 3, b = 1, θ|y ∼ Gamma(16, 6) Some prior and posterior summary statistics of θ are: E [θ ] = E [θ |y ] = a =3 b Mode(θ) =...
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## This note was uploaded on 02/22/2013 for the course MATH 604 taught by Professor Tadasee during the Spring '13 term at Georgetown.

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