## 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:

### computer science

##### Description

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.

## Related Questions in computer science category

##### Disclaimer
The ready solutions purchased from Library are already used solutions. Please do not submit them directly as it may lead to plagiarism. Once paid, the solution file download link will be sent to your provided email. Please either use them for learning purpose or re-write them in your own language. In case if you haven't get the email, do let us know via chat support.