Problem 1. Assume a set of i.i.d. samples of a Gaussian random variable X = {x1, x2, . . . , xN }, with
mean µ and variance σ
2
. Define also the quantities:

Show that if µ is considered to be known, a sufficient statistic for σ
2
is S¯
σ2 . Moreover, in the case where
both (µ, σ2
) are unknown, then a sufficient statistic is the pair (Sµ, Sσ2 ).

Problem 2. Let the observations xn, n = 1, 2, . . . , N, come from the uniform distribution

Problem 3. Assume that xn, n = 1, 2, . . . , N, are i.i.d. observations from a Gaussian N (µ, σ2
). Obtain
the MAP estimate of µ, if the prior follows the exponential distribution

Problem 4. Let x1, x2, . . . , xN be i.i.d. distributed according to the following Poisson distribution:

Problem 5. Maximum-likelihood methods apply to estimates of prior probabilities as well. Let samples
be drawn by successive, independent selections of a state of mature ωi with unknown probability P(ωi), i =
1, 2, . . . , M. Let zik = 1 if the state of nature for the kth sample is ωi and zik = 0 otherwise.

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