CMPUT466- Assignment 4 Solved

Problem 1.

Prove that, ifindependent.𝑃(𝑋, 𝑌) = 𝑓(𝑋)𝑔(𝑌) for some function 𝑓 on 𝑋only, and 𝑔on 𝑌only, then 𝑋 and 𝑌 are

𝑃don’t know how to start a proof, it’s a good indicator that the proof is simple. Just follow thedefinitions.Hint(𝑋): Useand 𝑃𝑃((𝑌𝑋),, use the definition of marginal𝑌) = 𝑃(𝑋)𝑃(𝑌) to prove independence,probability. The rule of the thumb is that, when wewhich is simply the definition. To obtain

Problem 2.

Prove that the expectation is a linear system.

𝔼𝑋∼𝑃(𝑋)[𝑎 𝑓(𝑋) + 𝑏 𝑔(𝑋)] = 𝑎 𝔼𝑋∼𝑃(𝑋)[𝑓(𝑋)] + 𝑏 𝔼𝑋∼𝑃(𝑋)[𝑔(𝑋)]

You may treat 𝑋 as a discrete variable.

Hint: Again, use the definition 𝔼𝑋 

Problems 3 and 4 concern the following setting.

  Letbe a continuous random variable uniformly distributed in the interval, where      are unknown parameters.

 

 We have a dataset, where each data sample is iid drawn from the above distribution, and we would like to estimate the parameters         and      .

Problem 3.

(a)  Give the likelihood of parameters.

(b)  Give the maximum likelihood estimation of parameters.

Problem 4.

(c)   Prove that MLE is biased in this case.

(d)  Prove that MLE is asymptotically unbiased if  .

Hint: A parameter estimation being biased is said in terms of the current dataset  , although we imagine that   could be repeatedly drawn from a repeatable trial. In other words, we do not assume   goes to infinity in (c). Being asymptotically unbiased means that, if   goes to infinity, the bias would become smaller and converge to 0.