Recent content by safina

May I ask what is needed to be computed in order to compare the estimators of the population mean \bar{y}_{N}, in terms of accuracy? That is to compare the multivariate ratio estimator \bar{y}_{RM} with the Y-only estimator \bar{y}_{n} in terms of accuracy.

Let X be a single observation from the density f\left(x;\theta\right) = \theta x^{\theta -1}I_{(0,1)} (x), where \theta > 0. What is the P\left[X \geq \frac{1}{2} |\theta > 1|\right]? Please help me structure the solution of this problem

Homework Statement The number of sales made by a used car salesman, per day, is a Poisson random variable with parameter \lambda. Given a random sample of the number of sales he made on n days, what is the most powerful test of the hypothesis Ho: p = 0.10 versus Ha: p = 0.25, where p is the...

P\left[\frac{X_{1}+X_{2}}{2} > 0.99\right] = \int^{1}_{0}\int^{0.98}_{0} 1 dx_{1}dx_{2} = 0.98 Is this right? If I double integrate the joint pdf with when X1 goes from 0 to 1 and X2 goes from 0 to 1, I got P\left[\frac{X_{1}+X_{2}}{2} > 0.99\right] = 1. I think it is not right because...

I really am confused how to calculate this probability. Also I think there is no table for uniform distribution.

the joint pdf of X_{1} and X_{2} is \frac{1}{\theta} \frac{1}{\theta} = \frac{1}{\theta^{2}} But I still don't get how is it related in getting \alpha

Homework Statement Given that X is a uniform random variable on the interval (0, \theta), we might test Ho: \theta = 1 versus the alternative H_{1}: \theta = 2 by taking a sample of 2 observations of X and rejecting Ho if \bar{X} > 0.99. Compute \alpha2. The attempt at a solution \alpha =...

It is defined that the population variance is S^{2}= \frac{1}{N-1}\sum^{N}_{1}\left(y_{i} - \bar{y}_{N}\right)^{2} or \sigma^{2}= \frac{1}{N}\sum^{N}_{1}\left(y_{i} - \bar{y}_{N}\right)^{2}. Also that the V\left[\bar{y}_{n}\right] = \frac{N-n}{N}\frac{S^{2}}{n} = \left(\frac{1}{n} -...

May I ask how to find the E\left(ln x\right) and Var\left(ln x)? The X_{i} are random sample from the f\left(x; \theta\right) = \theta x^{\theta - 1}I_{\left(0, 1\right)}\left(x\right) where \theta > 0. I need the information in finally solving the Cramer-Rao lower bound for the variance of...

Okey. Thank you statdad for your reply. But, I will be very happy if you will tell me how to find an unbiased estimator of this \kappa\left(\theta\right).

Oh, here's what I've done. \frac{nr}{\hat{\lambda}}= \sum x Solving for \lambda: \hat{\lambda} = \frac{rn}{\sum x} = \frac{rn}{n \bar{x}} = \frac{rn}{n \frac{r}{\lambda}} = \lambda Is this not right?

Homework Statement For a random sample X_{1}, ..., X_{n} from the Poisson distribution, find an unbiased estimator of \kappa\left(\theta\right) = \left(1 + \theta) e^{-\theta}. The Attempt at a Solution I know that the pmf of Poisson distribution is f\left(x; \theta\right) =...

Alright, thank you for all your replies. I've tried figuring them out. Here are the outcomes. Kindly check if these are right. \frac{d}{d\lambda} log L\left(\underline{y}; r, \lambda\right) = \frac{nr}{\lambda} - \sum y Equating the derivative above to zero results to...

Homework Statement Let X_{1}, ... , X_{n} be a random sample from f\left(x; \theta\right) = \theta x^{\theta - 1} I_{(0, 1)}\left(X\right), where \theta > 0. a. Find the maximum-likelihood estimator of \theta/\left(1 + \theta\right). b. Is there a function of \theta for which there...

Okay, thank for that. Can you help me further for the exact form of the likelihood function so that I can take the log on both sides afterwards?