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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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
12 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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5 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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10 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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11 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
27 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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19 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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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
12 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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8 views

Normal distribution using excel.

Exercise 1 Using Excel, evaluate P(Z<1.34236). Put your result in cell A10. I got 0.91026 Exercise 2 Find the table value T(1.34236) by looking at the table. Put the value you found in the ...
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11 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
20 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
51 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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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
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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8 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
31 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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8 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
18 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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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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9 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
39 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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26 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
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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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26 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 ...
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1answer
7 views

Construct 95% lower CI for proportion of adults who voted poor. Of 1060 adults, 54% voted poor environment.

I know that the lower CI will be theta - Z(alpha/2)*sigma. The value for Z is 1.96 because it is a 95% CI, but I am not sure how to find theta and sigma. The only other piece of information was that ...
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1answer
28 views

Ways to arrange ALGEBRA so AA occurs

So the permutations of this qould be 7!, and I know that there are 2 objects of type A, but how can we isolate the events where those objects occur consecutively?
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1answer
39 views

Follow-on question to “Fifty men and thirty woman…”

This questions relates to this question Fifty men and thirty woman are lined up at random. How do I find the expected number of men who have a woman standing next to them. and the answer given by ...
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1answer
48 views

There are 8 balls which appear identical. However, 1 is heavier than the rest. How do you find the ball with 2 weighings?

I understand there are similar problems but I am not sure how to go about constructing this problem with set of balls that are not exponents of 3^n. I know I need at least 2 weighings to find the ...
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1answer
26 views

probability question, two uniformly distributed independent events, what's the probability that a third event will occur? [duplicate]

let's say there's two events, a and b both a and b are uniformly distributed and have a range of [100,400] a and b are independent i know that the probability that a=A is 1/300 and the probability ...
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1answer
19 views

Data measured 1.225 times a year. Probability of measuring higher than X is 0.1. Probability of measuring a value higher than X in any given year? [on hold]

I´m stuck with a conversion issue from "Return period measured in amount of measurements" to "Return period in years". There are 1007 measurements for 822 years, or 1.225 measurements per year. All ...
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2answers
19 views

How do I weight votes based on number of possible voters?

Scenario: I have a book club that reads a book every month. My website allows the readers to give a 5 star rating to each book. We have 10 members and a rule that a member may only vote on a book they ...
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1answer
20 views

Frequency Distribution and Throughput

I am conducting an experiment on a couple of computer systems but the results I have don't make sense to me. I made each system perform 1000 operations: System A performs operations at a rate of ...
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10 views

Expectation Maximization question

I came across this question while practicing EM question but I don't understand how to apply EM in this scenario. What's the latent variable here? Is it the grade of each student? What will be the ...
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36 views

How do I determine if a statistical relationship exists in a real life problem: honor roll class assignments?

My question relates to Determine if being on honor roll gives an advantage in being assigned to the math class (I've seen similar questions involving hits on a webpage) which I believe should be ...
3
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1answer
31 views

Bivariate Probability question

Let $X$ be the time (in minutes) that John spends waiting for a bus on his way home from uni, and let$ Y$ be the time he spends waiting for a train. $X$ and $ Y$ have joint density ...
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2answers
23 views

Probability of a deck of cards [on hold]

In a deck of cards, what is the probability of drawing four kings? Not sure if it is rule of compliment?