Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

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Divergence of Chi-squared statistic

I want to write "proof" that a ${\chi}^2$ statistic becomes larger and larger as the sample size increases. I have come up with the following: For ${\chi}^2=\sum_{j=1}^{n} \frac{(O_j - ...
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12 views

Deriving marginal effects in a conditional multinomial logit model

I'm having difficulties deriving the derivative $\frac{\partial P_{ij}}{\partial x_{ih}}=-p_{ij}p_{ih}\beta$ when $P_{ij}=\frac{e^{x_{ij}^{'}}\beta}{\sum_{h=1}^{M}e^{x_{ij}^{'}\beta}}$ I've already ...
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23 views

What does this plot signify?

Well I am totally new to box plots and here is a boxplot I generated using Matlab for a certain software project. I am asked to tell the significance of the box plot, and well, I don't know what it ...
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1answer
21 views

Interpretation of confidence interval

Say I have a 95% confidence interval of the sample mean. Does that mean there is a 95% probability that the population mean lies within that interval?
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Identify at least three courses of variations and mention which of these you suspect to have larger contribution to the variation and why? [on hold]

An experiment is performed to compare the cleansing action of two detergents A and B. 10 clothes are washed with each of the detergents. All the 20 clothes washed are made from the same cotton cloth ...
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1answer
18 views

How to use lower bound to condition distribution

Say that I know that the height of a population comes from a distribution $P_H$. I now pick a random individual. However, someone tells me that he is taller than $H_m$. What is the new distribution ...
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1answer
23 views

How to test quality of probability estimates?

I have a Markov chain model which produces a probability distribution for absorption in 4 possible absorbing states. I.e. the model estimates the probability distribution for a discrete random ...
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2answers
33 views

Paired Observation Hypothesis Testing

I'm doing a project presentation for applied statistics about hypothesis testing. Me and my partner claimed that the average amount of money that people spend per month on groceries is less than or ...
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13 views

Estimating elasticity of y with respect to x in a log-log specification

The question My rudimentary workings so far is that; log(y_i/x_i) = log(y_i)-log(x_i) Factorise, so, log(y_i/x_i) = log(y_i) + upsilon_i - log(gamma_i + 1) Thus, elasticity of y to x is always >1 ...
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38 views

Expected Value of exponetial form

calculate expected value of $y$ when $$ f(y)=\frac{e^{\frac{y(1-\theta)}{\theta}}}{\theta} $$ I think $E(y)= \theta$ can I verify it? Thanks a lot!
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13 views

Distribution function of Wilcoxon signed-rank test

How would I determine the distribution function of the S+ test statistic for the Wilcoxon signed-rank test when N = 4?
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1answer
26 views

Probability proof and making sure I cannot make further simplifications to my answer

The question asks me to compute the probability (sums are ok) of the probability of having at least one of r cells empty with n>r balls thrown at the cells with equal likelihood of landing in any of ...
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1answer
14 views

Sample Variance Divided by Sample Size?

In my statistics book I found an example comparing paired samples. They computed the mean of the difference, $\bar{D} = 10.27$ and the sample standard deviation (or sample standard error) from the ...
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21 views

Finding MLE and UMVUE for $\theta$ for the following distribution??

I was working on following problem: Let $X_1,X_2,...,X_n$ be a random sample from a distribution with PDF given by $$f(x|\theta)=\theta^{-c}cx^{c-1}e^{-(\frac{x}{\theta})^c}$$ a) Find the MLE for ...
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16 views

What is this similarity formula? (Statistics)

Somebody gave me some code for finding similarity between two sets of data, and I was wondering if someone could help me identify the formula used? Here are the comments displaying the formula: ...
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1answer
17 views

Posterior density for θ - Gamma(α, β)

I want to find the posterior density for $\theta$ with a prior Gamma distribution $(\alpha, \beta)$. $$ f(x\mid\theta) = \frac 1 2 \theta e^{-\theta |x|}, \quad -\infty < x < \infty $$ From ...
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1answer
12 views

Finding minimal sufficient for the following distribution???

Hi I was working on finding the minimal sufficient for the following distribution $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}}I(0\le x\le 1),\,\,\,\,\,\,\theta>0$$ By factorization theorem ...
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11 views

Gaussian filtering

I'm reading a paper and don't get how they tackle the drift of a gaussian process. We are in the setting of isonormal Gaussian processes. Let $Z$ be a Gaussian process with covariance operator ...
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45 views

Incremental Bayes Rule

I want to use Bayes theorem on a problem multiple times. I am given the following information: $P(G) = 0.7$ $P(M) = 0.3$ $P(ET|G) = 0.3$ $P(ST|G) = 0.5$ $P(NT|G) = 0.4$ $P(ET|M) = 0.0$ ...
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1answer
24 views

Maximum likelihood estimator of an exponential distribution

I am working with the above function. I want to find the maximum likelihood estimator. I found the likelihood function for this distribution, by applying the likelihood function to every ...
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1answer
27 views

Conditional Probability of a group of Mice

There are six female and four male mice in a group of 10. One of the female mice and two of the male mice have a particular disease. Suppose that two mice are selected at random from the group without ...
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2answers
21 views

Is it possible to calculate the Xth percentile of a collection of wind vectors?

I have a collection of 30 weather model wind vectors (wind speed and direction) all valid at a specific point in space & time. It's fairly easy to break these apart and calculate a 75th percentile ...
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14 views

Mle of phone calls [closed]

Assume the number of emails received per minute by a service centre is a Poisson random variable with parameter lambda . If in 10 minutes 20 emails are received, find the MLE of lambda .
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1answer
22 views

Using Moment Generating Function to prove Z is standard normal

Suppose $X_1,...,X_n$ is a random sample from a normal distribution with an unknown mean $\mu$ ,known standard deviation $\sigma$ and sample population $\bar{X}$. Show (using moment generating ...
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56 views

Prediction Interval from Markov Chains

Thank you for taking the time to look at my question. Short, less involved question: How do you find Prediction Intervals with non-Gaussian residuals? Here is the situation: I have made a model that ...
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21 views

Hypothesis and Power Testing Poisson Random Variable

I have some problems trying to solve the following problem: Let $f_{0}$ and $f_{1}$ be pdfs for discrete integer-valued random variables, where $f_{0}(n) = \frac{1}{6} \mathbb{1} [n ∈ ...
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24 views

Expectation in Bayes rule

Let $L(\theta,\delta)=(\theta-\delta )^2e^{\frac{(\theta-100)^2}{900}}$ with $X\sim N(\theta,100)$ and $\theta\sim N(100,225)$ find the bayes rule. I already founded that posterior is ...
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1answer
32 views

MSE of uniform distribution

I would like to know how to calculate the MSE for a Uniform Distribution on (θ,2θ) I know that MSE is the variance of the method of moments estimator (mme). I have found this to be 3θ/2 Then I think ...
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1answer
26 views

What is the name of this statistical principal

I was wondering how I could shorten this explanation into fewer words. I am assuming their is a statistics term for this but I do not know it. Is there one? If I flip a coin 10 times and get 8 ...
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1answer
26 views

What kind of distribution is this [closed]

$$f(x,\theta) = \begin{cases}1/\theta,& \theta \leq x\leq 2\theta\\ 0,& \text{otherwise}\end{cases}$$ Is this a geometric distribution?
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1answer
23 views

Proof explanation - weak law of large numbers

Let $(X_i)$ be i.i.d. random variables with mean $\mu$ and finite variance. Then $$\dfrac{X_1 + \dots + X_n}{n} \to \mu \text{ weakly }$$ I have the proof here: What I don't understand is, why ...
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21 views

Is this the correct way of calculating the median?

I just completed the June $2009$ mei stats paper. On the mark scheme for question $5ii$ it shows the median being calculated using $600$, shouldn't it be $600+1$? I thought the median was calculated ...
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1answer
18 views

Maximum-Likelihood-estimator for number of marbles

Let there be n marbels in a box. Each has a unique number from 1 to n written on it. You pick a marble, write down the number and put it back. This process is repeated N times. Give a ...
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1answer
22 views

Linear Regression with Paired Data

For a sample of paired data (x,y), t tests are performed for the slopes of the population regression lines of y on x and of x on y. The null hypothesis in both tests is $H0:β=0$. Is it possible for ...
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30 views

Conditional probability for product of two statistics [closed]

For two independent positive statistics $X$ and $Y$ and $Z=XY$. Let $\tilde{Z}=\log_{10}XY-\left\lfloor{\log_{10} XY}\right \rfloor$ be the fractoinal part of $Z$. Is \begin{split} ...
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17 views

Normal Distribution and Construction of Independent Variable [closed]

Suppose $X$ and $Y$ are bivariate normal such that $E(X) = E(Y) = 0$ and $Var(X) = Var(Y) = 1, Cov(X,Y) = 1$ Then how can I find a non constant random variable which is a function of $X$ and $Y$ , ...
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1answer
27 views

Maximum Likelihood of single observation

I'm stumped on this problem... I keep getting an undefined answer of having to solve -20 = 0. The likelihood function I get is $e^{-20\alpha}$. So I have $y_i=$ $ \begin{cases} 1& ...
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1answer
31 views

excel linear transformation [closed]

Does anyone know how to do this linear transformation in excel? 1800 723 650 600 372 326 326 204 153 150 150 87 84 57 53.5 51 51 47 45 45 42 40 30.6 29.2 28.5 20 20 16 14 ...
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13 views

estimate on the sum of Rademacher functions

Let $(r_n)_{n\in\mathbb{N}}$ be Rademacher functions defined on $[0,1]$. See https://en.wikipedia.org/wiki/Rademacher_system for the definition of Rademacher functions. For any large integer $k$, ...
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15 views

Why is the mean of the $t_1$ distribution undefined? [duplicate]

The pdf of the $t_1$ distribution is symmetric about $0$, just like the pdfs of $t_k$ for $k>1$, so why does it not also have an expected value of 0?
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42 views

approximating a uniformly distributed random variable

Suppose that $U$ is a uniformly distributed (continuous) random variable on $[0,1]$. Let's say that I am interested in finding 3 discrete points $u_1,u_2,u_3$ which approximate $U$ in some sense. My ...
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1answer
26 views

t value question

You are conducting a study to see if students do better when they study all at once or in intervals. One group of 12 participants took a test after studying for one hour continuously. The other group ...
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1answer
23 views

Find approximation for size of population over time

Assume you start with a population of an objet of size $1$. Assume that a new objet of size $1$ is born at each date and that existing objects double in size in each period. Over time the sequence of ...
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1answer
16 views

Figuring out weighted criteria numbers/percentages?

I'm a bit confused on how I should go about accomplishing this/unsure of the terminology. So let's say I have a list of $10000$ voters. I want to get a composition that looks like: $34%$ Democrats ...
2
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1answer
44 views

sampling distribution question

Need clarification on a binomial sample example: we drew a sample of size $100$ from a binomial($m = 2$,$p = 0.2$) distribution and observed $76$ of the $x_i = 0$, $20$ of the $x_i = 1$ and $4$ of ...
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1answer
31 views

A machine produces memory sticks of varying lengths…

Full question: A machine produces memory sticks of varying lengths, distributed uniformly between 2 and 12 mm. Memory sticks longer than 10 mm do not meet the design criterion and must be scrapped. ...
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21 views

Moment generating function trouble understanding solution

Suppose that Y is a single observation from an exponential distribution with mean θ. Use the method of moment-generating functions to show that $\frac{2Y}\theta$ is a pivotal quantity and has a ...
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1answer
41 views

Average volume of set of cubes using the mean and variance of its side lengths.

First, i tried this question: The side lengths of a set of squares have an average of 5 and variance of 4. What is their average area? Let X = The Side Length From this question , I knew we had to ...
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18 views

Confused with this $G$ to find the maximum likelihood estimation!

Let $p_{ij}$ denote the probability that an observation falls into cell $(i,j)$ of a two-way contingency table containing $r$ rows and $c$ columns. A total of $x_{oo}$ observations are taken and ...
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Second Order Conditions and Maximum Likelihood Estimator for Normal and Exponential distributions

I can't seem to show the Second Order Condition for the MLE of the exponential distribution is <0. Does anyone have any hints? Same problem for the normal distribution when looking for MLE of the ...