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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13 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
12 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
27 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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0answers
14 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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13 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
9 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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17 views

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
20 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
17 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
26 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
8 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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0answers
16 views

Using an “auxiliary random experiment” to achieve a desired significance level

My question is somewhat simple, but, nonetheless I am not entirely convinced I am solving it correctly. I need to use the use the Neyman Pearson Lemma to test for $H_o : \theta = .5$ vs. $H_1 : ...
1
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1answer
17 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
5 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
16 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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0answers
22 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
12 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
28 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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24 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
8 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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0answers
7 views

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
7 views

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
2 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
15 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
14 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 ...
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0answers
23 views

Is it necessary to normalize likelihood within an event space before further multiplication among events?

Say I have observed data, and parameters $A,B$: Parameter $A$ contains possible values: $a_1,a_2,a_3$ Parameter $B$ contains possible values: $b_1,b_2,b_3$ Now, assume I know the likelihood of ...
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0answers
14 views

Data Significance. How do I analyse my data to find meaningful information? [on hold]

I have data on the time taken and CPU ticks taken for different cryptography algorithms to run. I used multiple size files, different key sizes and obviously different algorithms over multiple ...
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1answer
10 views

Variance with minimal MSE in normal distribution

Given $X_1,...,X_n$ ~ i.i.d. $N(\mu, \sigma^2)$ where the mean is unknown, let the estimator for $\sigma^2$ be $\hat{e} = p\sum_{i=1}^n(X_i-\overline{X})^2$ How do I choose $p$ so that this estimator ...
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1answer
27 views

Finding the variance of the time series defined as $x_t=\phi x_{t-1}+w_t$, for $t=2,3,4,…$.

Let $w_t$ be white noise with variance $\sigma_w^2$ and let $|\phi|<1$ be a constant. Consider the process $x_t=w_1$ and $x_t=\phi x_{t-1}+w_t$ for $t=2,3,...$. I need to find the variance. I ...
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1answer
35 views

Basic function manipulation and simplification question for $f((x-f(x))^2)$

I've run into a bit of a wall trying to understand why the following two equations are equivalent: $$f((x-f(x))^2) = f(x^2)-f(x)^2$$ I'm running into this with calculating population variance in ...
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0answers
11 views

Alternative single-letter notations in statistics

I'm learning about statistics, and a thing that is difficult for me is the common use of multi-lettered variables and functions. For instance Standard Error of $\beta$ is written $SE(\beta)$ and the ...
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10 views

How to decrypt a ciphertext by using the mutual index of coincidence?

I am trying to decrypt a Vigenére cipher text. I have found the key length by computing Index of Coincidence of substrings. The key length is 12. I know the letter frequencies the string and the ...
3
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1answer
52 views

Probability - Poisson arrival of rain

I'm trying to solve this Poisson problem. A rain shower lasts 10 minutes and in that time deposits $10^6$ raindrops over 100 $m^2$. a) What is the probability of at least one drop landing in 1 $cm^2$ ...
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0answers
17 views

What would be the sum of the deviations? [on hold]

Here the standard deviation is 0, the variance is 0 and the range is also 0. THere is no spread in the data. What is the one way this can happen?
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1answer
18 views

Statistics - Regression (The Linear Model) [on hold]

hey guys, I've been stuck on this proof for a couple hours now. Could someone help me out please? Thanks!
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2answers
25 views

Mutual information expressed as Kullback-Leibler divergence

My lecturer defines the mutual information: $$ I(X;Y\mid Z) = D_{KL}\big(p(X,Y\mid Z)\parallel p(X\mid Z)\;p(Y\mid Z)\big)$$ Is this correct? I feel like it doesn't really make sense to say that; ...
-2
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0answers
10 views

Multi-step probability problem. Noise and Stochastic Processes. [on hold]

Please see the image below! I am having issues with this problem and would love a solution.
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2answers
33 views

Is it appropriate to use $P(X=x^2)=P(X=x)$ in a secondary statistics textbook in this content?

Immediately got my attention is that where it wrote $\text{Note } P(X=x^2)=P(X=x).$ I just don't think it should be written like this, as this is wrong! Also, I think the following is wrong: ...
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1answer
13 views

What do the following exponential QQ plots tell me about the distribution of sample data?

I have two samples and these are their QQ exponential plots with line y=x through the origin overlayed. I am trying to deduce what the distributions of the two samples are. Do these suggest they ...
0
votes
1answer
31 views

Chebyshev's inequality for 1 standard deviation results in 0?

In applying Chebyshev's inequality to a probability distribution, the following is the given equation: $$p(\mu - c*\sigma \le X \le \mu + c*\sigma) \ge 1 - \frac{1}{c^2}$$ This indicates for any ...
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2answers
19 views

Covariance between $X$ and $Y$ of a bivariate normal distribution?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X$= 5 mL, $\sigma_Y$= 2 mL, $\mu_X$= 120 mL, $\mu_Y$= 100 mL, and $\rho$ = 0.6. How do I find the covariance of $X$ and $Y$? I know the ...
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0answers
16 views

Proof: Sum of two independent gaussian vectors is a gaussian vector

I want to show that the sum of two independent gaussian vectors is a gaussian vector. We had, that a gaussian vector can be written as $X=A*Z+b$ where $A$ is a real matrix, $b$ is a real vector and ...
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1answer
19 views

Econometrics/Statistics, variance and means

here's the problem I can't figure out on my own: The weight of a randomly selected student, (W), has a mean of $170$ and variance of $10$. Defining the new random variable ($Y$): the total weight of ...
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0answers
10 views

How to get the Riesz representative of the derivative of $L(K):=\text{tr}(\Lambda^* K A)$

$\DeclareMathOperator{\tr}{tr}K,\Lambda, A$ here are appropriate matrices. The question is not completely accurate as I can differentiate it, but I would prefer it to be in the form $⟨DL,h⟩$ for some ...
1
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1answer
40 views

Significance level for a hypothesis test for a linear regression

Consider linear regression model $Y_i=a+b\cdot x_i+\epsilon_i$, $i=1,2,3,4,5$, where $a,b\in\mathbb{R}$ are unknown and $x_1=x_2=1,x_3=3,x_4=x_5=5$, $\epsilon_i$ are iid, normally distributed with ...
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0answers
27 views

If B is a N(0,1) R.V., show $E[B^4] = 3$

I've read in Elementary Stochastic Processes by Mikosch (p. 98), that it is a well known fact that: If B is a N(0,1) R.V., $E[B^4] = 3$ I also see something equivalent (but uncited) on the ...
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1answer
87 views

Discrete distribution problem in medical application

im having trouble with question 2 on one of my math papers. I would greatly appreciate it if someone could help me out here, preferably give me worked out solutions for this question. Thank you for ...