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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stratified random sample

A report showed the number of homicides in each state. In Indiana, Ohio, and Kentucky, the number of homicides was, respectively, 380, 760, and 260. Suppose a stratified random sample with the ...
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16 views

Indepedent Continuous Random Variable Question - Showing that LHS = RHS

I don't understand this. I'm struggling, bit by bit, to understand some of this material (I've asked a few questions already on these topics) but it feels like a slow grind and I'd rather not wait ...
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1answer
18 views

Conditional probability distribution notation versus conditional probabilities of a single sample space?

When writing conditional marginal probabilities, the following seems to be the notation: $$p_{i|Y=y_{j}} = P(X=x_{i}|Y=y_{j}) = \frac{P(X=x_{i},Y=y_{j})}{P(Y=y_{j})}=\frac{p_{ij}}{p_{+j}}$$ This is ...
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1answer
22 views

which choice is better?

Lets say you were writing a program to play checkers. Im simplifying the numbers, but the gist should be obvious. Your program calculates the odds of Move A to have a 100 chances to win the game and ...
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1answer
15 views

X and Y Joint Density Question - Compute the Density of X and Y

I simply cannot understand this. My TAs aren't available until tomorrow and I really do not want to put this off until then. I'd like to have some idea of how to do this beforehand. The question is: ...
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1answer
10 views

Order statistics when variables have different distributions

Let a,b and c be random variables, where a~U[0,8] and b~U[3,8]. Let c=max{a,b}, What is the mean of c? In general, let $(x_1, ..., x_n)$ be independently distributed in different supports. What is ...
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3answers
18 views

Algebraic simplification of likelihood ratio

Can someone help me understand how this: ...
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1answer
16 views

If $nE\|X_n-Y_n\|^2=o(1)$, why $\sqrt{n}(X_n-Y_n)=o_p(1)$ [on hold]

How to show that if we have $nE\|X_n-Y_n\|^2=o(1)$, then $\sqrt{n}(X_n-Y_n)=o_p(1)$? Here $X_n,Y_n$ are two random elements.
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1answer
28 views

Lift-club rates (This should be really easy)

Right, this is actually a real-life problem. I want to join Bob and Joe's lift club. Joe usually pays about \$40 a week (in total) to drive between A and B (for fuel). (Driving from A to B and back is ...
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0answers
11 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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1answer
14 views

How to measure reaction time when given a right or wrong answer test? [on hold]

Here is an example of some data. test 1 test 2 test 3 correct 20 10 13 wrong 4 8 12 Lets ...
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0answers
9 views

Invariance of MLE

Random sample has pdf:$f(x|\theta)$ Then the MLE of $\theta$ is $$\hat{\theta}=\arg\max_{ \theta}\sum_{i=1}^n\log f(x_i|\theta)$$ $\tau=g(\theta)$ My question is why $\tau$ 's MLE is ...
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0answers
11 views

Is the Laplace Transform of the convolution power the product of the Laplace Transformed convolution?

In statistics, the definition of $F^k$ is the k-fold convolution of $F$ with itself, where $F$ is some common distribution. I am wondering if the following holds, if: $$ L_{F^{k}(x)} = ...
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2answers
46 views

How to verify how good my data is?

I'm in an undergrad physics lab right now and I have taken some data. The theoretical curve should be proportional to $\cos^2(\theta)$. How can I quantify how close my data values are to this ...
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0answers
18 views

Wilson score interval proof / derivation [on hold]

I need to get from here to here Can anyone help me?
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0answers
17 views

Lower std.deviation of a distribution?

How would one lower the std.dev => $\sigma$ for an arbitrary distribution? The reason why I ask, is because I have a distribution which tend to get far above my threshold value, and i am running out ...
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0answers
9 views

Sampling Distribution help [on hold]

A sample of size 13 is drawn from a normally distributed population with known standard deviation 1.30. The sample mean is 4.25. Part A Find the lower and upper limits of a 95% confidence interval ...
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1answer
34 views

What distribution it is based on the histogram? [on hold]

I generated this histogram in r and was trying to determine which distribution I should use, my guess is normal or Binormial. But I'm not sure, can anyone help please?
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1answer
22 views

Setting up the expected value for $x_t=\sin(2\pi U t)$.

We have the series $x_t=\sin(2\pi U t)$ where $t=1,2,3,\ldots$ and $U$ is uniform on the interval $(0,1)$. I have to find the expected value of $x_t$. I always thought that if $X$ is a continuous ...
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1answer
10 views

In hypothesis testing..

... we use two tail test and divide the area by $\frac{\alpha}{2}$ each. Why don't we divide it into some other propotion say $\frac{3}{4} \alpha$ and $\frac{\alpha}{4}$? Here $\alpha$ means type 1 ...
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1answer
30 views

Given a survival rate matrix, describe what can be said about it

Given this matrix equation: $$\begin{bmatrix} c_{k+1} \\ t_{k+1} \\ a_{k+1} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0.33 \\ 0.18 ...
2
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0answers
23 views

Double Integral of an Exponential Function with an Absolute Value in the Numerator of the Exponent

This is a question related to statistics, but my major concern relates to the setup and evaluation of integrals. So I decided this question was better suited for Mathematics Exchange than CV. I know ...
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0answers
17 views

Mathematical statistics: Pollen dispersal directionality

What Information am I looking for? Think about a tree that is sending pollen all over the place. Because of wind, most pollen grain will go toward one direction. Imagine, we split the 2D area around ...
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1answer
22 views

What does SSB and SSW of ANOVA tell you about your data?

This might be a stupid question. I know how to calculate them but mine not sure what they are telling me about my data set. What does it mean if $\sum $$(SSB)^2$ $\ge \sum (SSW)^2$, or vise versa, ...
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1answer
16 views

Sampling distribution question with unknown n.

Suppose that 53% of the population of voters were in favor of fighting the global warming. If we wanted to conduct a random sample of size $n$ of voters, how many should I survey if I want the ...
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3answers
28 views

Finding the probability of an event with binomial distribution using a normal approximation

A Tarheels basketball player is obsessed about practicing his free throws. It is known that he is $75\%$ free throw shooter. One morning he decides to shoot $100$ free throws. You may assume that ...
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1answer
26 views

Calculate mean and std.dev from a lot of coordinates

How do i calculate the mean coordinate and the standard deviation of a cloud of (x,y) coordinates. I know how to calculate the mean, but i am struggling with calculating the std. deviation.
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0answers
16 views

Using one way ANOVA on table with two column (correct and false)

I've used a one way ANOVA on test data for 4 tables. Each table has 3 columns, participant ID, correct and false. The participants took a test where they had to get as many the correct answers as they ...
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0answers
16 views

the significance of these two successive bumps [on hold]

Suppose there is a sin curve with amplitude $A$, which is $I=A\sin(t)$ . If we detect two small bumps near two successive peaks of the $\sin$ curve, one with a small offset $t_1$, the other with ...
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1answer
10 views

R nls function and starting values

I'm wondering how I can find/choose the starting values for the nls function as I'm getting errors with any I put in. I also want to confirm that I can actually use the nls function with my data set ...
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0answers
16 views

Under what assumptions is the following first moment monotone?

I'm working on an economic model and am encountering the following mathematical issue. Let $x\sim \mathcal{N}(\mu,1)$, and define $$V(\mu)=\int_0^{\hat x(\mu)}x ...
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Statistical Distributions [on hold]

How to solve: If $X$ is a normally distributed random variable with mean $\mu = 80$ and standard deviation $σ = 15$, what is the probability that the mean $\overline{X}$ ̅of a random sample of size ...
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1answer
21 views

Find the 90% confidence interval for the population proportion. [on hold]

In a survey of 8000 women, 5431 say they change their nail polish once a week. Construct a 90% confidence interval for the population proportion of women who change their nail polish once a week.
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1answer
20 views

Find the 95% confidence interval and interpret the results [on hold]

A random sample of 38 200-meter swims has a mean of 3.96 minutes and the population standard deviation is 0.06 minutes. Construct a 95% confidence interval for the population mean time. Interpret the ...
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0answers
30 views

Optimal bandwidth for histogram density estimator [on hold]

Derive optimal bandwidth for the histogram density estimator and estimate error bounds with respect to AMISE (asymptotic mean integrate square error). I tried using this code in Matlab but doesnt ...
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0answers
9 views

Intuition for the formulas of mode and median for grouped data

To estimate the mode and median of a grouped data set, in my Statistics class they presented these formulas to me: Mode = L + [(fm-f1) / (fm-f1)+(fm-f2)] x h where: L is the lower class boundary of ...
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1answer
18 views

Covariance and Correlation in Multinormal random variable

Find the covariance and correlation of $N_i$ and $N_j$, where $N_1, N_2, \ldots,N_r$ are multinormal random variable. At the beginning, I think that I have: ...
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0answers
6 views

Normal probability plot convert formula: $f_i$= (i - 0.375)/(n + 0.25)

$f_i$ = (i - 0.375)/(n + 0.25) I've seen this formula around the web that is used when trying to convert exponential density function to a normal probability plot. What is the significance of 0.375 ...
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1answer
17 views

Reason behind convergence in probability definition

A sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon > 0$ $$\lim_{n\to\infty}\Pr\big(|X_n-X| > \epsilon\big) = 0$$ But ...
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0answers
7 views

Generalized linear model with Weibull response

I want to perform a nonlinear regression to a dataset where the response variable seems to have a Weibull distribution (I performed Kolmogorov-Smirnov test to check this hypothesis). However, this ...
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1answer
33 views

Kolmogorov-Smirnov two-sample test

I want to test if two samples are drawn from the same distribution. I generated two random arrays and used a python function to derive the KS statistic $D$ and the two-tailed p-value $P$: ...
2
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0answers
14 views

when is the maximum likelihood estimator measurable

For a random variable $X$, a class of probability measures $P_\theta$ for $\theta\in \Theta$ and their densities $f_{\theta}$ w.r.t. some common measure $\mu$, we can define the maximum likelihood ...
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1answer
29 views

Expectations of squared sum question

I can't seem to figure out why these expectations turn out the way they do, I am currently studying about the Fisher Information. If $X_1,X_2,...,X_n $ are all iid Poission($\lambda$) , then going ...
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0answers
26 views

OLS: Estimation with negative coefficients

I have probably an easy problem, however I'm not really sure how to do it: Basically, I would like to estimate a linear regression with OLS. So far so easy. However, the model that I would like to ...
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0answers
10 views

Interpretation of sufficient statistic in the continuous case

A statistic $S = S (X)$ is called sufficient for $\theta$ if there is a $P_{X \mid S} (\cdot \mid s)$ that doesn't depend on $\theta$. So if $S(X)$ is a discrete random variable and we know $S (X) = ...
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Can I use Geometric Distribution to find the law of a total?

I have a variable X which is the amount of minerals in a dL(deciliter) of water. X follows a Normal Distribution X~N(μ,σ). I have the probabilty of the P(a ≤X< b) in a dl, where a and b are ...
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0answers
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Kernel estimate in boundary point

Good moorning, I wonder how to prove that if $X_{1}, \ldots, X_{n}$ are iid from exponential distribution with expected value 1, then the expected value of its kernel density estimator in zero is ...
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0answers
3 views

Variance of Inhomogenous Poisson process in a given window

Consider some variable $X\sim \operatorname{Poi}(\lambda(t))$ to be Poisson-distributed with some parameter $\lambda$ dependent on time, where we know how the random variable $\lambda$ is distributed. ...
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1answer
18 views

Finding binomial probability, bernoulli trials

The following table lists World Series Lengths for the fifty years from $1926$ to $1975$. Test at the $0.10$ level whether these data are compatible with the model that each World Series game is an ...
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0answers
15 views

Chebyshev's Theorem Sample Size

I'm working on a problem and getting two different results. It states that E(Xi)=0 and that Var(Xi)=3 for each response in a survey sample of weights. I'm trying to find the minimum sample size to ...