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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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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13 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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11 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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15 views

the significance of these two successive bumps

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
9 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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8 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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24 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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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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10 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 : ...
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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
15 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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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
21 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 $P$: ...
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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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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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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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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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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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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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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
25 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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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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9 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 ...
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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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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
23 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; ...
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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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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 ...
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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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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 ...
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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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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
85 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 ...
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0answers
27 views

Estimating a probability

I am interested in the probability of a random deal in bridge to be a par- zero-deal (a deal where no player can make any contract assuming perfect play with all hands visible) The events I need ...
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26 views

Data analysis: How did people beat the Great Hall game?

This is the game: There is a Great Hall with 102 doors. 100 of these doors lead to one of 100 different side rooms. The 101st door, at the end of the Great Hall leads to the Great Tower, where ...