9 Why is this estimator biased? 8 Find UMVUE of $\frac{1}{\theta}$ where $f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$ 8 Simple examples of uncorrelated but not independent $X$ and $Y$ 7 Sufficient statistics for Uniform $(-\theta,\theta)$ 7 Prove that $\text{Corr}(X^2,Y^2)=\rho^2$ where $X,Y$ are jointly $N(0,1)$ variables with correlation $\rho$

### Reputation (6,887)

 +10 Unbiased estimator of binomial parameter +10 Is a minimal sufficient statistic also a complete statistic +10 Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$? +10 Finding credible interval for $\mu_1-\mu_2$ when $X_i\sim N(\mu_1,\sigma^2)$ and $Y_i\sim N(\mu_2,\sigma^2)$

### Questions (42)

 18 Distribution of $\frac{\sum_{i=1}^n X_iY_i}{\sum_{i=1}^n X_i^2}$ where $X_i,Y_i$s are i.i.d Normal variables 13 $(2Y-1)\sqrt X\sim\mathcal N(0,1)$ when $X\sim\chi^2_{n-1}$ and $Y\sim\text{Beta}\left(\frac{n}{2}-1,\frac{n}{2}-1\right)$ independently 12 Constructing example showing $\mathbb{E}(X^{-1})=(\mathbb{E}(X))^{-1}$ 12 If $X$ and $Y$ are independent Normal variables each with mean zero, then $\frac{XY}{\sqrt{X^2+Y^2}}$ is also a Normal variable 11 Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$?

### Tags (128)

 163 self-study × 97 41 inference × 25 68 mathematical-statistics × 55 37 umvue × 17 65 distributions × 50 33 hypothesis-testing × 25 57 normal-distribution × 30 31 probability × 20 52 estimation × 28 29 sufficient-statistics × 18

### Bookmarks (673)

 1136 Making sense of principal component analysis, eigenvectors & eigenvalues 560 What is the difference between “likelihood” and “probability”? 491 What is the intuition behind beta distribution? 491 What is the difference between test set and validation set? 471 Why square the difference instead of taking the absolute value in standard deviation?