Let Y₁, Y2, ..., Y₁ denote a random sample from the probability density function 0 0. Show that Y is a consistent estimator of 0/(0 + 1). @ye-1, f (y) = = {0,

The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…

Let X1,...,X, be independent random variables with pdfs -{* 0 -i(0 – 1) 0. Find a two-dimensional sufficient statistic for 0.

Let X1, X2,...,X, be a random sample from a distribution with density function e if x > 0 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?

A variable X has a probability function f(x, 0) = 0e 0x; x≥0 y >0. The parameter 0 is estimated using two different estimators of simple random samples of size two: 01 2x+4x2 and 2=4x+5x2 3 3 Select the best estimator for \theta by comparing the Mean Squared Errors

Let  X be a continuous random variable with range  [−ln5,0]  and its probability density function is given by the following function: f(x)=ce^−x, where  cc is a constant.  (1) Find the value of  c . Answer:   (2) Find the probability  P(−ln2≤X≤0) . Answer:

A machine has a useful life of 4 to 9 years, and its life (in years) has a probability density function defined by f(x) = 1 1 Find the probability that the useful life of such a machine selected 15 at random will be longer than 7 years?