Questions tagged [statistics]
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.
2,335
questions
7
votes
2
answers
13k
views
Bivariate Normal Conditional Variance
I am given the parameters for a bivariate normal distribution ($\mu_x, \mu_y, \sigma_x, \sigma_y,$ and $\rho$). How would I go about finding the Var($Y|X=x$)? I was able to find E[$Y|X=x$] by writing $...
- 1,109
4
votes
2
answers
8k
views
Covariance of order statistics (uniform case)
Let $X_1, \ldots, X_n$ be uniformly distributed on $[0,1]$ and $X_{(1)}, ..., X_{(n)}$ the corresponding order statistic. I want to calculate $Cov(X_{(j)}, X_{(k)})$ for $j, k \in \{1, \ldots, n\}$.
...
- 1,717
4
votes
2
answers
23k
views
Proof that if $Z$ is standard normal, then $Z^2$ is distributed Chi-Square (1).
Suppose that $Z\sim N(0,1)$ and let $V=Z^2$. Prove that $V\sim \chi^2(1)$.
I want to use the method of moment generating functions, because I already understand the proof using the method of ...
- 1,821
3
votes
2
answers
5k
views
Cramer Rao lower bound in Cauchy distribution
I need to calculate the Cramer Rao lower bound of variance for the parameter $\theta$ of the distribution $$f(x)=\frac{1}{\pi(1+(x-\theta)^2)}$$
How do I proceed I have calculated $$4 E\frac{(X-\...
- 2,632
3
votes
2
answers
2k
views
Use Rao-Blackwell Theorem to find the UMVUE
Suppose that $X_1,X_2,...,X_n$ is a random sample from a normal distribution, $X_i\sim N(\mu,9)$.
Find the UMVUE (uniformly minimum variance unbiased estimator) of $P(X\le c)$ where $c$ is a known ...
- 2,365
1
vote
1
answer
3k
views
Maximum likelihood when usual procedure doesn't work
I am trying to get the maximum likelihood estimate for the parameter $p$. The distribution is the following:
$$ f(x\mid p) = \begin{cases}
\frac{p}{x^2} &\text{for} \ p\leq x < \infty \
\\ 0 ...
- 536
1
vote
2
answers
4k
views
Is $\bar X$ a minimum variance unbiased estimator of $\theta$ in an exponential distribution?
I proceeded by finding $\operatorname E(\bar X).$ I considered $\bar X$ as a constant and simply got the term itself. This should suggest that $\bar X$ is not an unbiased estimator of $\theta$. That's ...
0
votes
1
answer
7k
views
Find the maximum likelihood estimator for $\theta$ when $f(x)=2\theta^{-2}x, 0\leq x \leq \theta$
Find the maximum likelihood estimator for $\theta$ when $f(x)=\frac{2x}{\theta^2}, 0\leq x \leq \theta$.
This should be a really easy question but I somehow cannot seem to get the right answer. My ...
- 3,379
122
votes
4
answers
183k
views
What is the difference and relationship between the binomial and Bernoulli distributions?
How should I understand the difference or relationship between binomial and Bernoulli distribution?
- 2,772
49
votes
11
answers
22k
views
Why does Benford's Law (or Zipf's Law) hold?
Both Benford's Law (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and Zipf's Law (given a corpus of natural ...
- 9,944
37
votes
2
answers
172k
views
maximum estimator method more known as MLE of a uniform distribution [closed]
Let $ X_1, ... X_n $ a sample of independent random variables with uniform distribution $(0,$$
\theta
$$
) $
Find a $ $$
\widehat\theta
$$
$ estimator for theta using the maximun estimator ...
- 3,063
31
votes
5
answers
57k
views
Correlation between three variables question
I was asked this question regarding correlation recently, and although it seems intuitive, I still haven't worked out the answer satisfactorily. I hope you can help me out with this seemingly simple ...
- 435
26
votes
3
answers
12k
views
Expected value of applying the sigmoid function to a normal distribution
Short version:
I would like to calculate the expected value if you apply the sigmoid function $\frac{1}{1+e^{-x}}$ to a normal distribution with expected value $\mu$ and standard deviation $\sigma$.
...
- 263
19
votes
1
answer
20k
views
MLE for Uniform $(0,\theta)$
I am a bit confused about the derivation of MLE of Uniform$(0,\theta)$.
I understand that $L(\theta)={\theta}^{-n}$ is a decreasing function and to find the MLE we want to maximize the likelihood ...
- 5,175
18
votes
7
answers
106k
views
Proof variance of Geometric Distribution
I have a Geometric Distribution, where the stochastic variable $X$ represents the number of failures before the first success.
The distribution function is $P(X=x) = q^x p$ for $x=0,1,2,\ldots$ and $...
- 323