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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Probability of error in estimation

We are given the cumulative partition function of the rainy days in march and july. We have a paper, without the name of the month (but we know it is one of these two), on which 14 rainy days are ...
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38 views

Validity of conditional CDF proof via PDF integral

Given the question: $$\text{Show that}\ F_X(x\mid A) = \dfrac{\Pr(A\mid X\leq x)}{\Pr(A)}\cdot F_X(x)$$ I have seen the solution via probabilities 'directly'. My question is whether the following ...
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1answer
18 views

Test of confidence intervals?

In one of my assignments I have to "test" if the confidence intervals for a set of parameters in a mixed effect model is accurate. I'm asked to simulate from fittet parameters and there after refit ...
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22 views

Is the following data set possible?

"Is it possible to create a data set where $\bar{x}=30.0$, range $R=10$, and variance $s^2=40.0$?" I feel sort of dumb asking this question, but I'm not quite sure I'm on the right track. I know that ...
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Binormal Probability [closed]

Q. At a certain time, Hawaii contained 770,000 people, 60% Asian, 39% White and 1% Black, if a random sample of 7 persons was drawn. What are the chance that a majority are Asians? Can someone ...
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24 views

Is the event is plausible or not?

An atlete specialized in long jump events jumps an average of $\bar x=7.91m$ in $12$ trials. The standard error of the mean jump distance in these trials is $0.2m$. Is it plausible that when the ...
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1k views

Precise mathematical translation of the 68–95–99.7 rule?(Not a proof!)

The rule: In statistics, the 68–95–99.7 rule, also known as the three-sigma rule or empirical rule, states that nearly all values lie within 3 standard deviations of the mean in a normal ...
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12 views

Correlated random effects?

Say I have a linear model where two random effects are possibly correlated, do I still have a random effect matrix for each of them? for instance if $Z_1X=U_1,Z_2W=U_2$ are two possibly correlated ...
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Sampling With Replacement No-Repeats

I am about to do a survey on a population of 250 individuals. I will be performing a Two Sample T Test For A Difference Between Two Populations but am having trouble meeting the conditions for the 10% ...
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15 views

Trying to find the specific name for such a function.

We know that if $\frac{f(x + t)}{f(x)}$ increases in $x$ for $x > 0$ and $t > 0$, then $f$ is said to be a $\log$ convex function. But if we have $\frac{f(xz)}{f(x)}$ increasing in $x$ for $x ...
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16 views

Using Poisson Distribution Method

Q) The question is as follows: suppose that a book of 200 pages contains 20 printing mistakes.Assume that are errors are randomly distributed throughout the book and x , the number of errors per page ...
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367 views

Variance of a MLE $\sigma^2$ estimator; how to calculate

Let $\mathbb X_1;X_2;...;X_n$ be an i.i.d. random sample from N~(0, $\sigma^{2}$). a. Find the variance of $\sigma^{2}_{MLE}$ So I found $\sigma^{2}_{MLE}$ by taking the derivative of the log of ...
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35 views

Probability of being highest OR second highest draw from a distribution

In my situation, I have a distribution F(x) over some compact interval. Say I take $n$ iid draws from the distribution. I want to find the probability that one draw, $x_i$, is the highest of the $n$ ...
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2answers
1k views

Same mean, different standard deviation in data sets

How would a data set containing the values of a variable with a mean of 50 and a standard deviation of 3 compare with another data set containing the same variable, but a mean of 50 and a standard ...
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1answer
974 views

Predicting the number of orders from future customers

Tamara is reviewing recent orders at her deli to determine which meats she should order. She found that of 1,000 orders, 450 customers ordered turkey, 375 customers ordered ham and 250 customers ...
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3answers
443 views

Confidence interval for Uniform($\theta$, $\theta + a$)

I am encountering a difficulty with the following task. Have I made a mistake, or is this an inherent flaw in the notion of confidence intervals? (Other such flaws exist.) Consider a random sample ...
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1answer
30 views

MLE of Integer Valued Normal Distribution

If Z is a normal random variable on $\mathbb{R}^d$ with parameters $(\mu,\Sigma)$ and we know that $\mu\in \mathbb{Z}^d$ and $\Sigma \in \mathbb{Z}^{d+}$; then how can we solve this MLE problem for ...
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33 views

Is there a simpler calculation for the probability of exactly x occurrences over n trials?

What I've worked out for this, where x is the specific number of occurrences, n is the number of trials and p is the probability of an occurrence during a single trial, is this: \begin{equation}‎ ...
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862 views

Median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
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23 views

Probability and statistics Hypothesis testing

I'm not sure if this is the right place to publish this question, but I am trying to understand something about hypothesis testing. recently in a lecture on the subject a professor emphasized how ...
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1answer
408 views

Weighted Standard Deviation for Histogram Bin Height

I'm plotting some binned data in the form of a histogram. Say I have 10 data points, each composed of a bin to be placed in, and then a "height". Then I might have something like: Bin Height 0 - ...
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10 views

Allan Deviation Explanation

I'm not sure if this is the correct stackexchange for this but I'm looking for an explanation of Allan Deviation or Allan Variance. I'm trying to learn it for research I'm working in for analyzing ...
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3answers
421 views

Variance of a max function

Say $x_1$ and $x_2$ are normal random variables with known means and standard deviations and $C$ is a constant. If $y = \max(x_1,x_2,C)$, what is $\mathrm{Var}(y)$? Well, I forgot to tell that $x_1$ ...
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1answer
1k views

Mana Maximization (Hearthstone)

I recently started playing Hearthstone and a statistic / probability question came up my mind. Here's a quick breakdown: The game is a turn-based card game which involves "points" that you can used ...
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1answer
8 views

What can we say about the concentration around 0 linear transformation of Gaussian random variables?

I have a matrix $X \in \mathbb{R}^{n \times m}$ such that each $A_{ij}$ is a Gaussian with mean $0$ and variance $1$. We have $m > n$. I also have a vector $v \in \mathbb{R}^m$ such that $||v||_2 ...
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Chances in combinatorics [closed]

There are 124 apples, you and 9 more people are taking them. What are the chances for you to take the one that you wanted?
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30 views

Probability Normal Disitrubution, P(X<=1.5)

I think this is simple but I'm not sure how it works (I probably just dozed off while reading this part). The important info is that X is a normal random variable with mean 3 and standard deviation ...
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1answer
9 views

Shift graph towards mean with time?

Say I have a bar graph with one dependent variable and one independent variable. Given a time $t$, I essentially want to modify the given graph so that as $t -> \infty$, all the bars become equal ...
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2answers
64 views

Probability of dice with a cumulative successes

Needing some Math help! I'm working out part of a system for a game I'm working on, and I need to know if I need to tweak some things, so I'm checking to see if it all adds up (no pun intended). ...
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1answer
37 views

Expected value of norm of multivariate normal distribution random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this ...
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35 views

Expansion of reciprocal of quadratic

Can I expand $\frac{1}{1-.7B-.3B^2}$ into an infinite series? Where B is the backwards operator in time series. I was thinking $\frac{1}{1-(-.3B)}\frac{1}{1-B}$. Express this as a product of a ...
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Is it possible to find a standard deviation for a sample with only its average or mean available?

I have four groups each with a sample size of 84 and I want to run a t-test on them but I have no clue what their standard deviations are. Is there any way to find the standard deviations or do I have ...
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541 views

Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms…

Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms. Suppose 25 of these resistors are randomly selected to be used in a circuit. a) What is the ...
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22 views

equivalence pdf for the difference of n normal distribution functions

I know about the equivalence pdf for the sum of n normal distribution functions, is which their convolution. Could you please explain the pdf for the case of subtracting n normal distributions?... ...
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1answer
32 views

How to find the probability of an estimator?

I'm working on some homework and am having a hard time finding the probability of an estimator. The question says that $Y_i$ ~ Bernoulli($p_2$), where $p_2 = 0.1$. The estimator $p_2 = \bar{Y} = ...
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21 views

CV Percentage Error with Confidence Variable

I am trying to calculate a confidence variable (CV %) of two numbers where the numbers them selves have a confidence range. Appologies for my sloppy representation, I am somewhat of an equation novice ...
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distribution of non-central chi random sample

Suppose that $X_1,X_2, \ldots, X_n$ is a random sample from a non-central chi distribution with $1$ degree of freedom. What is the distribution of the sample variance of $X_1,X_2, \ldots, X_n$?
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Working out closed form of shifted poisson distribution

In the article "Bayesian variable selection for Poisson regression with underreported responses" the author defines $t_i^0$ as the number of actual occurences in a study in the $i$th covariate ...
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1answer
27 views

Prove a result in multiple linear regression

This arises in multiple linear regression. Given $m, n \in \mathbb{N}$ and matrices $X \in \mathbb{R}^{m \times (n+1)} (m > n + 1), H = X(X'X)^{-1}X' \in \mathbb{R}^{m\times m}, I = I_m$ and $J ...
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1answer
83 views

Prove that $E(\mathbf{u}|\mathbf{X})=\mathbf{0}$ implies $Cov(\mathbf{x},\mathbf{u})=\mathbf{0}$

Let \begin{equation} \mathbf{y}=\mathbf{X}\mathbf{\beta}+\mathbf{u} \end{equation} where $\mathbf{y}=\begin{bmatrix}y_1 \\ \vdots \\ y_n\end{bmatrix}$, $\mathbf{X}=\begin{bmatrix}X_{11} & ...
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1answer
36 views

Combining iterations to come up with a final solution

For a game I'm building, I need to know how many tries it will take for a player to roll several specific numbers (say 1-30 out of 100 possible numbers). The numbers can be repeated and only need to ...
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16 views

A sequence of random variables converge in distribution to some random variable

Given a sequence of i.i.d $Uniform[0,1]$ random variables, $X_1, X_2,...$. By taking logs find a random variable $X$ such that $(X_1...X_n)^{\frac{1}{\sqrt{n}}}e^{\sqrt{n}} \to X$ in distribution. I ...
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2answers
199 views

Using sufficiency to prove and disprove completeness of a distribution

Let $X_1, \dots ,X_n$ be a random sample of size $n$ from the continuous distribution with pdf $f_X(x\mid\theta) = \dfrac{2\theta^2}{x^3} I(x)_{(\theta;\infty)}$ where $\theta \in \Theta = (0, ...
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621 views

Show a statistic is not sufficient

Let $T$ be a sufficient statistic. Suppose $f(T)$ is not a one-to-one function of $T$. Show $f(T)$ is not a sufficient statistic.   I think this should be proved by contradiction. Since $f$ is not ...
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Not complete but minimal sufficient statistic

Let $X =(X_1,\ldots, X_n), ~ X_i \mathrm{iid} \sim \mathcal N ( \theta, \theta^2), ~ \theta \in \Theta = \mathbb R \setminus \{0\}, ~ T(X)=(\sum_{i=1}^n X_i, \sum_{i=1}^n X_i^2)$. I figured out that ...
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11 views

Generate feature vector matrix given mean and covariance matrix

What should be the steps i should follow for the following question using MATLAB generate matrix X which has 400 columns of feature vectors, first 200 columns from class 1, then 200 from class 2. ...
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1answer
94 views

Why the chi-squared statistic follows chi-squared distribution?

The formula for the Chi-Square test statistic is the following: $$\chi^2=\sum_{i=1}^n\frac{({O_i-E_i})^2}{E_i}$$ where $O_i$ is observed data, and $E_i$ is expected. I am just curious why this ...
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16 views

Histogram with skewed data

I have a set of data which I want to plot a histogram of to get a feel for which bins contain the most items. There are 200,000 data points which range from 0.08 to 5,000. Problem is I know the data ...
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17 views

Martingale Dozen 1/3 statistics.

When playing on dozens there is 12/37 chance to win, and the return is 1/3. To turn odds to your favour you have to double up your money for a chance of winning money. So if i lose on my 1 unit bet, i ...
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Test $ H_{0}: b = 0.5 $ vs. $ H_{1}: b < 0.5 $ at the $ 5 \% $-level for a sample size of $ n = 100 $.

I have data with density $$ f(x;b) = \begin{cases} \dfrac{2 x}{b} & \text{if $ 0 < x < b $}, \\ \dfrac{2 (x - 1)}{b - 1} & \text{if $ b < x < 1 $}. \end{cases} $$ I have ...