Suppose X is a two-class mixture model, given as

where Z ∼ Bernoulli(p).

(a) Write down the marginal density function fX(x).

(b) State the law of iterated expectation, and use it to fifind the expectation of X.

(c) State the law of total variance, and use it to fifind the variance of X.

 2.  

The continuous random variable X has probability density function

fX(x|θ) = θxΘ–1                                 0 < x < 1, θ > 0.

A random sample (X1  , X2, . . . , Xn) is used to estimate the parameter θ.

(a) Defifine what it means for a random variable to be continuous.

(b) Determine the cumulative distribution of X.

(c) If θ is known, explain how a random sample from U ∼ Uniform (0, 1) could be used to produce a random sample with distribution X, proving any general results that you use.

(d) Compute the method of moments estimator of θ.

(e) Compute the maximum likelihood estimator of θ. (Remember to check explicitly that your estimator does indeed maximize the likelihood.)

(f) Let Y denote the method of moments estimator of θ. In the case n = 1, show that the sampling distribution of Y has density