Suppose that Y_1,Y_2,Y_3 denote a random sample from anexponential distribution with density function f(y)=(\frac{1}{\lambda})\cdot e^{\frac{-y}{\theta}},y>0 =0 elsewhere Consider the following five estimators of \theta: \hat{\theta}_1=Y_1,\hat{\theta}_2=\frac{Y_1+Y_2}{2},\hat{\theta}_3=\frac{Y_1+2Y_2}{3},\hat{\theta}_4=\overline{Y} a) Which of these estimators are unbiased? b) Among the unbiased estimators, which has the smallest variance?

Question
Sampling distributions
Suppose that $$\displaystyle{Y}_{{1}},{Y}_{{2}},{Y}_{{3}}$$ denote a random sample from anexponential distribution with density function $$\displaystyle{f{{\left({y}\right)}}}={\left({\frac{{{1}}}{{\lambda}}}\right)}\cdot{e}^{{{\frac{{-{y}}}{{\theta}}}}},{y}{>}{0}$$
=0 elsewhere
Consider the following five estimators of $$\displaystyle\theta$$:
$$\displaystyle\hat{{\theta}}_{{1}}={Y}_{{1}},\hat{{\theta}}_{{2}}={\frac{{{Y}_{{1}}+{Y}_{{2}}}}{{{2}}}},\hat{{\theta}}_{{3}}={\frac{{{Y}_{{1}}+{2}{Y}_{{2}}}}{{{3}}}},\hat{{\theta}}_{{4}}=\overline{{{Y}}}$$
a) Which of these estimators are unbiased?
b) Among the unbiased estimators, which has the smallest variance?

2021-02-03

.

Relevant Questions

Two random variables X and Y with joint density function given by:
$$f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & elsewhere\end{cases}$$
Find the marginal density of X.
Two random variables X and Y with joint density function given by:
$$f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & elsewhere\end{cases}$$
Find the marginal density of Y.
Two random variables X and Y with joint density function given by:
$$f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & elsewhere\end{cases}$$
Find $$P(x<\frac{1}{3})$$
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
Assume that X and Y are jointly continuous random variables with joint probability density function given by
$$f(x,y)=\begin{cases}\frac{1}{36}(3x-xy+4y)\ if\ 0 < x < 2\ and\ 1 < y < 3\\0\ \ \ \ \ othrewise\end{cases}$$
Find the marginal density functions for X and Y .
Random variables $$X_{1},X_{2},...,X_{n}$$ are independent and identically distributed. 0 is a parameter of their distribution.
If $$X_{1}, X_{2},...,X_{n}$$ are Normally distributed with unknown mean 0 and standard deviation 1, then $$\overline{X} \sim N(\frac{0,1}{n})$$. Use this result to obtain a pivotal function of X and 0.
The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
$$\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}$$
where A is the cross-sectional area of the vehicle and $$\displaystyle{C}_{{d}}$$ is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity $$\displaystyle\vec{{{v}}}$$. Find the power dissipated by form drag.
Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, $$\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}$$, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
Let $$X_{1},X_{2},...,X_{6}$$ be an i.i.d. random sample where each $$X_{i}$$ is a continuous random variable with probability density function
$$f(x)=e^{-(x-0)}, x>0$$
Find the probability density function for $$X_{6}$$.
$$f(x,y)=f(x,y)=\begin{cases}cx^{2}e^{\frac{-y}{3}} \& 1<x<6, 2<y<4 = 0\\0 & otherwise\end{cases}$$
Draw the integration boundaries and write the integration only for $$P(X+Y\leq 6)$$
Suppose that $$X_{1}, X_{2} and X_{3}$$ are three independent random variables with the same distribution as X.
What is the ecpected value of the sum $$X_{1}+X_{2}+X_{3}$$? The product $$X_{1}X_{2}X_{3}$$?
Suppose a discrete random variable X assumes the value $$\frac{3}{2}$$ with probability 0.5 and assumes the value $$\frac{1}{2}$$ with probability 0.5.