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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Adding and Subtracting Normal Distributions

I think I know how to do this, but I'm not sure. I'm just hoping to check myself here before I do a bunch of work incorrectly. Suppose you have three independent normal distributions: Distribution A: ...
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Which function will fit this curve best?

I am trying to do a test of normality on this data set here. My QQ Plot looks like this . It looked like an arctan function to me. So my idea was to do a reverse "tan" function transformation on it. ...
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Analytic solution for matrix factorization using alternating least squares

The standard form for ridge regression aims to minimize the following cost function. $$ \min\ \ \sum_i(y_i-x_i^T\beta)^2 + \lambda\sum_j\beta^2_j $$ As described here, it's possible to differentiate ...
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Density function after a transformation.

Assume $Y$ has the density function $q(y) = 3y^2$ when $y \in (0,1)$, overwhise zero. Let $Z = -log(Y)$. Then; $$P(Z \le z) = P(-log(Y) \le z) = 1 - P(Y \le e^{-z}) = 1 - F_Y (e^{-z})$$ Then I ...
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How to do sampling for the following problem. [on hold]

There are 200 students with a mathematics exam marks. According to marks students are divided into five categories 0-20,20-40,40-60,... and I want to choose two random sample with 25 for a group. ...
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1answer
21 views

Assumptions of a probability distribution

Let $X$ be a continuous real-valued random variable indicating the fragility of a firm. Suppose that the firm defaults if $X$ takes a value above a threshold $u>0$. Hence $$ Prob(X>u) $$ is the ...
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Moment generating function of a uniform random variable.

My first attempt to this question was to find the first few moments about the mean and try to rearranging the those moments to obtain the general function as desired. However, when I tried to ...
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Finding variance with method of moments

In part (a), I know Var(Ni)=p(p-1) but how do I find the variance of the estimator with this result? And for part (b) of this question, I have to clue on how to tackle this question. I don't get ...
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24 views

Sufficient conditions for monotonicity with probability distributions

Let $X_i$ be a continuous non-negative real-valued random variable and $i=1,...,n$. $X_i$ are not necessarily independent over $i$. Let $b>0$, $\delta>0$. Consider $$ ...
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What distribution would describe this?

I start with 100 eggs, 10 of them being broken. I randomly select eggs without replacement until they are all split into baskets of 10 eggs each. Here's what I know: Best case scenario all 10 bad ...
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1answer
17 views

Statistics; Normal distribution question

I'm not sure if I am solving this question correctly A used-car dealership has found the length of time before a major repair is required on car it sells is normally distributed. Witha mean = 10 ...
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How do I find percentiles of data sets (Even vs odd)?

Given the following data set with an even number of values: $100, 100, 105, 113, 129, 132, 146, 152, 176, 200$ The value representing the 30th percentile, using the formula n(p/100) where n = sample ...
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24 views

Does Binomial variables independence implies Bernoulli variables independence

$X$, $Y$ are independent variables with Binomial distribution. $X={\sum_{i=1}^nX_i}$, $Y={\sum_{i=1}^nY_i}$. $X_i$, ($1\le i\le n$) are independent Bernoulli variables. Same applies for $Y_i$ Is the ...
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1answer
35 views

Probability of a point from one normal distribution being higher than a point from another independent normal distribution

Given two independent normal distributions: Distribution 1: Mean $= 23.95$, SD $= 7.44$ Distribution 2: Mean $= 16.29$, SD $= 7.79$ How often on average will a point from Distribution 2 be greater ...
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1answer
17 views

Finding method of moments

Question: In order to solve find the method of moments estimator, I know I need to first find the expected value of Y. But after finding $E(Y)= \frac{\theta}{2}$, what should I do next?
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Confidence level using unknown median

This is a general question. Lets say we are giving a sample space n. The sample space has an unknown median The order statistics of each person, $**X_{i}**$ in sample space is measured (lets say ...
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1answer
25 views

Calculating the probability of winning roulette after x bets

I'm going through all of my homeworks to study for my final and I'm getting hung up on this one problem I never figured out... A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you ...
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21 views

standard deviation and adjusted R-squared for simultaneous regressions

I am conducting a study that requires two steps of statistical estimation. First, I run a regular OLS regression, from which I gather three outputs that I need: coefficient values standard ...
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27 views

Statistical research [on hold]

I'm preparing for an exam and I came across this question in a book. Below, I understand that we have to assumed a split plot in time analysis with 3 way factorial with 3 drug $\times$ 2 sex $\times$ ...
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46 views

Statistical research question [on hold]

I'm preparing for an exam and I came across this question in a book. An experiment is done to examine ways to detect phlebitis during the intravenous administration of a particular drug. Phlebitis ...
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1answer
36 views

When using the Central Limit Theorem, how to scale the mean and variance depending on the number of samples?

So I'm reviewing my notes for the central limit theorem for my final and I'm getting hung up on one detail. The two questions below both utilize the central limit theorem, but they use it in ...
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26 views

case deletion formula restricted least square estimator [on hold]

hi any one please help me to find out a case deletion formula for restricted least square estimator? $$ \hat\beta = (X' X)^{-1} X'y-(X' X)^{-1} R' [R' (X' X)^{-1} R]^{-1} R(X' X)^{-1} X'y $$ i need a ...
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How to show U1 and U2 are independent (Stats)

Prompt: Assume that $Y_1, Y_2, Y_3$ and $Y_4$ are independently and identically distributed $N(\mu,\sigma^2)$ random variables. Show that $Y_1 + Y_2 – Y_3 – Y_4$ and $Y_1 – Y_2 + Y_3 – Y_4$ are ...
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explain how this confidence interval is correct [on hold]

An srs of $16$ households is selected in Houston and the number of remote controls is counted. we are interested in a $99\%$ confidence interval for the population mean number. In the sample, the mean ...
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Markov chain steady state existence

Is it possible for a Markov chain to have no steady state solution ? What is an example of such system ?
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Stats probability addition rule, multination rule

The directions are to calculate the following probability based on drawing cards without replacement from a standard deck of 52. What is the probability of drawing a 2 or a king on the first draw and ...
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2answers
28 views

Defining median for discrete distribution

In probability theory, a median of a probability distribution is a number $M$ such that the CDF of this distribution $F_\xi(x)$ satisfies $F_\xi(M)=\frac{1}{2} \tag1$ This works for continuous ...
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1answer
34 views

Number of samples needed to get a given expected distance

Suppose I have a surface in $\mathbb{R}^3$ with surface area $A$. How many points do I need to (uniformly at random) sample so that the expected distance from each point to its nearest neighbor is ...
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2answers
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Standard deviation of $Z = 9 - 3Y^3$

If Y is a random variable with probability mass function: Y | Pr(Y = y) -1 | 0.4 0 | 0.5 1 | 0.1 I need to find the standard deviation of $Z = 9 - 3Y^3$. ...
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1answer
39 views

Approximate using the central limit theorem

This is a homework question in a class I'm TAing, and I just wanted to make sure I'm not doing anything wrong, before I conclude for sure that the book has an error. Suppose $S_n$ is binomially ...
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1answer
24 views

Poisson Distribution and sample mean

I am currently working on some error analysis homework and I am having trouble understanding some basic concepts. In particular I don't understand if there is any difference between the sample mean, ...
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1answer
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Normal approximation of Poisson Distribution

Hi currently studying for a final exam and I just want to confirm my approach/answers to this problem are correct: Suppose that $X \sim \mathrm{Poisson}$. We wish to test $H_0: \lambda = 50$ vs ...
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inequality for real-valued Gaussian sums

I saw the following Lemma in an article: Let $\mathbf{b}\in \mathbb{R}^N$ be fixed, and let $\mathbf{\epsilon}\in \mathbb{R}^N$ be a random vector whose N entries are i.i.d. random variables drawn ...
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1answer
21 views

Mean and Variance of probabilities [on hold]

For a certain commodity which you buy, you can make either a $500$ profit with probability $0.5$ when you sell it, or $200$ with probability $0.3$ or lose $100$ with probability $0.2$. a. Find the ...
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Confusion with QQ plots

I've got some difficulty understanding QQ plots and the specific slope and intercept of the line which the data will approximate if it really is generated from the distribution we're comparing with. ...
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34 views

The efficiency property of estimator for second moment. [on hold]

Please help to improve the efficiency property of estimation for second moment. Statistical population is normally distributed. Sorry for my bad english, if something is wrong. Thanks in advance.
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1answer
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Can I sum variances to a compound variance?

Say I have three locations A,B,C and I have a person going from A to B and measure the time it takes. Same for B to C. Let the variance of the time it takes for the path AB be a and for BC b. Is it ...
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1answer
35 views

adjusted R squared with multiple dependent varialbles

A question about regression in statistics. What is the formula for adjusted R squared if there are multiple dependent variables
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Estimating the mode

I am interested in the following problem and after searching I am surprised I can't find anything useful about it. Consider a multiset $A= \{a_1,\dots, a_n\}$ of integers. Say you sample elements of ...
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1answer
27 views

Probability Density Function given X

I have this problem I am unable to solve in my book. It has a provided solution and I am unable to come to this conclusion. The problem is as follows: Suppose that a random variable X has a PDF given ...
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How to calculate p-value using an algorithm on R assuming that distribution is unknown.

For a given sample x, where x is distributed with a normal distribution mean known but variance unknown. I am testing the hypothesis that variance is equal to one or greater than 1. The question is to ...
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Montmort's card matching problem: Distribution of the number of matching cards?

(Introduction to Probability, Blitzstein and Nwang) Recall de Montmort’s matching problem from Chapter 1: in a deck of n cards labeled 1 through n, a match occurs when the number on the card ...
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1answer
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Let $X \sim \text{HGeom}(w,b,n)$, what is the distribution of $n-X$?

Let $X \sim \text{HGeom}(w,b,n)$, what is the distribution of $n-X$? The distribution of $X$ (e.g., number of white ($w$) balls in a sample of size $n$) is hypergeometric, so $$P(X=x) = ...
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Using gamma distribution to find the average duration of breaks after 10 calls with exponential distribution

Worker works 8 hours a day. Time between $ 2$ calls has $\exp(4)$ distrubution (expecting $4$ calls per hour). Duration of calls is $0$ (he just registers them). After $10$ calls he goes to $15$ ...
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1answer
23 views

The autocorrelation function - the result in the form of a vector.

I've implemented the autocorrelation function in Python according to the normalized autocovariance function for discrete signals, i.e: ...
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Huber's distribution to minimize the Fisher information (generalize to multivariate case?)

The question is about Robust Statistics (by P.Huber). Any suggestion will be appreciated, thanks. It is proved in Huber's book (Robust Statistics) that the optimal solution of the problem ...
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How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise.

As the title says, I want to estimate the variances needed for a Kalman filter from real sensor measurements only. For example we can take a temperature sensor, but the solution shall be as ...
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Expected value of variance and variance of variance?

From a certain system dynamical system I am able to calculate the expected error $\mathbb{E}(e_k)$ and the variance $\sigma^2 = \operatorname{var}(e_k) = \mathbb{E}(e_k^2) - \mathbb{E}(e_k)^2$, as ...
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How to change 1D Metropolis into 2D?

I've written a MATLAB function to generate random numbers from a given univariate distribution using the Metorpolis algorithm. Here it is: ...
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Using the $\chi ^2$ goodness of fit test to see if data is normal.

I have a data sample of size $n$ that I suspect comes from a normal distribution with some parameters $\mu, \sigma$. I wish to check this hypothesis using a $\chi ^2$ g.o.f. test with, say, $\alpha = ...