User StubbornAtom

19 votes

1 answer

829 views

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

asked Nov 15, 2018 at 17:51

14 votes

1 answer

524 views

Is $\frac1{n+1}\sum_{i=1}^n(X_i-\overline X)^2$ an admissible estimator for $\sigma^2$?

asked May 5, 2021 at 19:08

14 votes

2 answers

3k views

Constructing example showing $\mathbb{E}(X^{-1})=(\mathbb{E}(X))^{-1}$

asked Sep 30, 2017 at 17:59

13 votes

5 answers

661 views

$(2Y-1)\sqrt X\sim\mathcal N(0,1)$ when $X\sim\chi^2_{n-1}$ and $Y\sim\text{Beta}\left(\frac{n}{2}-1,\frac{n}{2}-1\right)$ independently

asked Feb 8, 2018 at 11:37

12 votes

1 answer

2k views

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

asked Mar 13, 2018 at 20:32

12 votes

1 answer

382 views

Validity of confidence interval for $\rho$ when $X\sim N_3(0,\Sigma)$ with $\Sigma_{ij}=\rho^{|i-j|}$

asked Mar 31, 2019 at 12:08

11 votes

1 answer

784 views

Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$?

asked Apr 26, 2019 at 10:01

11 votes

2 answers

3k views

On the existence of UMVUE and choice of estimator of $\theta$ in $\mathcal N(\theta,\theta^2)$ population

asked Jun 27, 2018 at 15:09

11 votes

3 answers

2k views

Distribution of $\sqrt{X^2+Y^2}$ when $X,Y$ are independent $U(0,1)$ variables

asked Jan 17, 2018 at 20:22

10 votes

2 answers

1k views

Showing $\frac{2X}{1-X^2}$ is standard Cauchy when $X$ is standard Cauchy

asked Dec 17, 2017 at 13:42

10 votes

1 answer

5k views

Definition of sample space

asked Oct 25, 2016 at 17:53

10 votes

2 answers

4k views

UMVUE of $\frac{\theta}{1+\theta}$ while sampling from $\text{Beta}(\theta,1)$ population

asked May 31, 2018 at 6:40

10 votes

2 answers

501 views

How to show that $\{N(\theta,1):\theta \in \Omega\}$ is not a complete family of distributions when $\Omega$ is finite?

asked Apr 5 at 22:28

8 votes

3 answers

467 views

Estimating $\theta$ based on censored data when $X_i\sim \text{Uniform}(0,\theta)$ with $\theta\ge 1$

asked Jan 27, 2022 at 13:46

8 votes

1 answer

277 views

Finding the distribution of sample range for a Beta population

asked Nov 6, 2018 at 15:13

8 votes

1 answer

332 views

Showing $Z_i$'s are independent if $Z_1=\sum_{i=1}^{n+1}X_i$ and $Z_i=\frac{X_i}{\sum_{j=1}^iX_j}, i\ge2$ when $X_i\sim\text{G}(\alpha,p_i)$

asked Dec 9, 2017 at 14:48

8 votes

1 answer

453 views

Distribution of $X+Y$ when $X$ and $Y$ are i.i.d with pdf $f(x)=\alpha\beta^{-\alpha}x^{\alpha-1}\mathbf1_{0<x<\beta}$

asked Apr 14, 2018 at 8:35

7 votes

2 answers

312 views

Correlation between an observation and its rank in a random sample

asked Nov 8, 2018 at 13:42

7 votes

1 answer

131 views

Are $U=\frac{2X_1^2}{(X_2+X_3)^2}$ and $V=\frac{2(X_2-X_3)^2}{2X_1^2+(X_2+X_3)^2}$ independent?

asked Apr 11, 2022 at 17:11

6 votes

3 answers

135 views

Is $P(|X_1|>k)\le P(|X_2|> k)$ when $X_i\sim N(\mu_i,\sigma^2)$ and $|\mu_2| \ge |\mu_1|$?

asked Jul 4, 2021 at 8:24

6 votes

2 answers

504 views

How to justify that $(Y_1,Y_2)$ is not bivariate normal without finding its exact distribution?

asked Dec 25, 2018 at 6:59

6 votes

1 answer

1k views

Distribution of $\sum_{j=1}^n\ln\left(\frac{X_{(j)}}{X_{(1)}}\right)$ when $X_i$'s are i.i.d Pareto variables

asked May 12, 2018 at 20:05

6 votes

1 answer

334 views

Is $\theta$ a location or a scale parameter in the $\mathcal N(\theta,\theta)$ and $\mathcal N(\theta,\theta^2)$ densities?

asked May 28, 2018 at 12:15

6 votes

2 answers

588 views

For independent RVs $X_1,X_2,X_3$, does $X_1+X_2\stackrel{d}{=}X_1+X_3$ imply $X_2\stackrel{d}{=}X_3$?

asked Sep 17, 2017 at 6:55

5 votes

2 answers

706 views

PMF of the number of trials required for two successive heads

asked Sep 19, 2017 at 17:04

5 votes

1 answer

6k views

Is it okay to write the square of expectation of a random variable $X$ as $\mathbb{E}^2(X)$?

asked Jun 24, 2016 at 11:30

5 votes

3 answers

4k views

Practical applications of the Laplace and Cauchy distributions

asked Aug 12, 2016 at 7:41

5 votes

1 answer

765 views

Showing MSE of $\bar{X}\mathbf1_{\bar{X}>0}$ is less than that of $\bar X$ when sampling from $\mathcal N(\theta,1)$ population

asked May 21, 2018 at 15:25

5 votes

1 answer

123 views

Upper bound for the probability $P\left[\left|\frac{Y_n}{n}-p^2\right|>\varepsilon\right]$

asked Jul 22, 2018 at 14:12

5 votes

1 answer

744 views

How to show that $X_{(1)}-\frac1n$ is the unique minimax estimator of $\theta$?

asked Feb 5, 2021 at 13:13

Stack Exchange Network

59 Questions

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

Is $\frac1{n+1}\sum_{i=1}^n(X_i-\overline X)^2$ an admissible estimator for $\sigma^2$?

Constructing example showing $\mathbb{E}(X^{-1})=(\mathbb{E}(X))^{-1}$

$(2Y-1)\sqrt X\sim\mathcal N(0,1)$ when $X\sim\chi^2_{n-1}$ and $Y\sim\text{Beta}\left(\frac{n}{2}-1,\frac{n}{2}-1\right)$ independently

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

Validity of confidence interval for $\rho$ when $X\sim N_3(0,\Sigma)$ with $\Sigma_{ij}=\rho^{|i-j|}$

Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$?

On the existence of UMVUE and choice of estimator of $\theta$ in $\mathcal N(\theta,\theta^2)$ population

Distribution of $\sqrt{X^2+Y^2}$ when $X,Y$ are independent $U(0,1)$ variables

Showing $\frac{2X}{1-X^2}$ is standard Cauchy when $X$ is standard Cauchy

Definition of sample space

UMVUE of $\frac{\theta}{1+\theta}$ while sampling from $\text{Beta}(\theta,1)$ population

How to show that $\{N(\theta,1):\theta \in \Omega\}$ is not a complete family of distributions when $\Omega$ is finite?

Estimating $\theta$ based on censored data when $X_i\sim \text{Uniform}(0,\theta)$ with $\theta\ge 1$

Finding the distribution of sample range for a Beta population

Showing $Z_i$'s are independent if $Z_1=\sum_{i=1}^{n+1}X_i$ and $Z_i=\frac{X_i}{\sum_{j=1}^iX_j}, i\ge2$ when $X_i\sim\text{G}(\alpha,p_i)$

Distribution of $X+Y$ when $X$ and $Y$ are i.i.d with pdf $f(x)=\alpha\beta^{-\alpha}x^{\alpha-1}\mathbf1_{0<x<\beta}$

Correlation between an observation and its rank in a random sample

Are $U=\frac{2X_1^2}{(X_2+X_3)^2}$ and $V=\frac{2(X_2-X_3)^2}{2X_1^2+(X_2+X_3)^2}$ independent?

Is $P(|X_1|>k)\le P(|X_2|> k)$ when $X_i\sim N(\mu_i,\sigma^2)$ and $|\mu_2| \ge |\mu_1|$?

How to justify that $(Y_1,Y_2)$ is not bivariate normal without finding its exact distribution?

Distribution of $\sum_{j=1}^n\ln\left(\frac{X_{(j)}}{X_{(1)}}\right)$ when $X_i$'s are i.i.d Pareto variables

Is $\theta$ a location or a scale parameter in the $\mathcal N(\theta,\theta)$ and $\mathcal N(\theta,\theta^2)$ densities?

For independent RVs $X_1,X_2,X_3$, does $X_1+X_2\stackrel{d}{=}X_1+X_3$ imply $X_2\stackrel{d}{=}X_3$?

PMF of the number of trials required for two successive heads

Is it okay to write the square of expectation of a random variable $X$ as $\mathbb{E}^2(X)$?

Practical applications of the Laplace and Cauchy distributions

Showing MSE of $\bar{X}\mathbf1_{\bar{X}>0}$ is less than that of $\bar X$ when sampling from $\mathcal N(\theta,1)$ population

Upper bound for the probability $P\left[\left|\frac{Y_n}{n}-p^2\right|>\varepsilon\right]$

How to show that $X_{(1)}-\frac1n$ is the unique minimax estimator of $\theta$?