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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Will someone explain this polynomial regression equation?

I am in high school and I need to write a program that does polynomial regression to any degree on a set of data for a personal project. I think that this Wikipedia Article has the equation that I ...
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72 views

Sum of residuals proof

Show that: $\sum x_i e_i=0$ and also show that $\sum\hat{y}_i e_i=0$. Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved ...
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67 views

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$ i) Find the regions where the joint pdf of $(Z,W)$ is positive. ii) Find the ...
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21 views

Covering deficits with values with different weights

SO I have a couple of assessments with specific weights as follows: Assignment 1: 5% => Mark 60% Assignment 2: 5% => Mark 53% Assignment 3: 5% Assignment 4: 5% Test 1: 30% => 47% Test 2: 30% ...
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12 views

Bounds of a Bivariate Function

I am given that $h(x, y) = \frac{x}{(x+y)}$ , $x > 0$ , and $y > 0$. I am supposed to deduce that the bounds for $h(x, y)$ are $0 < h(x, y) < 1$, but I do not understand how to arrive at ...
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46 views

Summation notation with ambiguous subscripts

I'm reading a paper which has the following description; Say we have a time series of correlated sequential observations of the random variable $X$ denoted $\{x_n\}_{n=1}^N$ from a stationary, time ...
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79 views

Please help me solve average problems

The least and greatest numbers in a list of 7 real numbers are 2 and 20, respectively. The median of the list is 6, and the number 3 occurs most often in the list. Which of the following could be the ...
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37 views

Algorithm to generate simulated data for any deterministic relation

We have a statistical model (M) that we want to benchmark against simulated data; our argument is that the model M is more expressive than other models in terms of being able to capture all kinds of ...
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35 views

Figuring out the distribution of sample variance

If I have a random sample $X_1,...,X_m$ with normal observations where mean $\mu$ and variance $\sigma^2$ then how can I show that $s_x^2=\sum_{i=1}^{m} ...
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29 views

Determining which test statistic is better under two different scenarios

Problem: I have 9 independent normal observations from a normal dist. with $\sigma=10$. And we want to test $H_0:\mu=150$vs $H_A:\mu\neq 150$. Also $\alpha=0.05$ which uses the test statistic ...
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25 views

Asymptotic test statistic for exponentially distributed data

I need an idea for tackling the following problem: Let $X_1,...,X_n\sim\mathrm{Exp}(\lambda)$ be an iid sample. We want to test $H_0:\lambda=\lambda_0$ vs. $H_1:\lambda\not=\lambda_0$ for some ...
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33 views

3 students share a room in a dormitory. they have 4 cups, 5 saucers, and 3 teaspoons,

3 students share a room in a dormitory. They have $4$ cups, $5$ saucers, and $3$ teaspoons, all different. In how many ways can they set the table for tea? (each set consists of a cup, saucers, and a ...
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92 views

Asymptotic Maxwell MLE distribution

Consider i.i.d. random samples $X_1,...,X_n$ from the Maxwell Density: $$ f_\theta(x)=\sqrt{\frac{2}{\pi}}\dfrac{x^2}{\theta^3}e^{-\frac{x^2}{2\theta^2}}I_{(0,\infty)}(x) $$ with $\theta > 0$. ...
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38 views

Is there any known method to fit plane onto sampling data?

For example I have the variables x, y (or higher dimensional data in general) and a probability density distribution p(x,y). I want to approximate p(x,y) as a linear function, a plane in this case, at ...
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689 views

Example of a real-world situation where multivariate analysis is applicable.

I have searched a lot of site to understand the situation where multivariate analysis is applicable. But not got any easily understandable example. Would you please give me a real-world example where ...
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394 views

Bayes estimator from a geometric distribution with a uniform prior

X is a random variable with Ber(p), 0 Y is the number of trials until a success occurs. Assume the prior p is unif(0,1). I have trouble in figuring out the posterior density f(p|Y). With the ...
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34 views

computing expectation of two arm bandit

assume you have a two arm bandit with one arm having a fixed, known probability of payoff $p = 0.6$ and another having an unknown payoff $q$, which is drawn uniformly from $[0,1]$. Each game the ...
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29 views

How to Calculate Population from A given Set Of Sampels

I have a sample set of data collected using a SRS of books with IDs from 1 to 100. {90,60,6,39,46,26,16} Using this data how can I estimate the max, in this senario I know the max id, but what if I ...
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83 views

estimate coefficients of $y = \alpha x + \beta y + \gamma z + \epsilon$

I know how to find $m$ and $b$ for $y= mx +b$, which is : $m= \frac{\bar{x}\bar{y}- \bar{xy}}{(\bar{x})^2 - \bar{x^2}}$ and $b= \bar{y} - m\bar{x}$ How can we estimate $\alpha, \beta, \gamma$ and ...
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86 views

Adding a constant to a $T$ distribution: does it preserve the sample variance and sample size?

Question is as stated: If $T_1$ follows a $T$ distribution with sample variance $s$ and sample size $n$ and $T_2 = T_1 + k$, does $T_2$ follow a $T$ distribution with mean $\mathbb{E}[T_1] + k$ ...
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34 views

How to confirm/disprove a hypotesis

I have a list of quadruples in the following form (name, smokes, mother-smokes, father-smokes) (Andrew, Y, N, Y) (Jessica, N, N, N) ... and I would need to ...
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23 views

Determining the statistical power of a test — is my approach correct?

Let $(X_1, X_2, X_3)$ be a sample taken from $\mathcal{N}(\mu, \sigma^2 = 12.2^2)$ (mean unknown, variance known). We want to test the following hypotheses: $$ H_0 : \mu = 0 \\ H_a : \mu < 0 ...
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93 views

Likelihood Ratio and Neyman-Pearson factorization theorem

I'm looking at a family of distributions given by $P = \{P_{\theta} \quad | \quad \theta \in \{0,1\} \}$. I'm trying to prove that $$T(x) = \frac{p_{1}(x)}{p_{0}(x)}$$ (i. e. the likelihood ratio) ...
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24 views

Get Progress from Three Values

I have a starting weight, current weight and target weight. How can I get a percentage of progress? Thanks!
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63 views

How to define the 'error'

I have true data $G$ and wrong data $F$. Both are $n$ dimension vector. $G\in \{G_i| 0<G_i<255\}, i = 1:n$. Because the ...
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2answers
64 views

What is my probability space and measurable space?

I have the following difference equation $$ \tilde{u}_k = \begin{cases} u_k & \text{if $\gamma_k = 1$, no signal lost} \\ \tilde{u}_{k-1} & \text{if $\gamma_k = 0$, signal lost} \end{cases} ...
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31 views

Suppose $40\%$ of the population possess a given characteristic … What is the probability $44\%$ or fewer possess the characteristic?

I have the following question: Suppose $40\%$ of the population possess a given characteristic. If a random sample of size $300$ is drawn from the population, then the probability that $44\%$ or ...
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1answer
33 views

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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1answer
50 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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96 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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46 views

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
54 views

Exponential(1) distribution of Normally distributed X and Y

Let $X_1,X_2,X_3,X_4,X_5$ be a random sample from the uniform pdf: $f(x)= 1$, $0<x<1$ zero otherwise. Show that $\ln X_i$ has Exponential($1$) distribution for $i=1,2,3,4,5$. Solution: Let ...
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1answer
22 views

formula for desired sample size

The following is a description of the standard deviation of the sample mean. $$\sigma(\bar X)=\sqrt {V(\bar X)}=\sqrt{\frac {N-n}{N-1}*\frac {\sigma^2}{n}}\leq D$$ Where $D$ is "a constant which ...
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1answer
29 views

confidence intervals for expected spending

A random sample of 10 motorists buying petrol are found to spend an average of £58.30 with estimated standard error £5.25.  Calculate a 95% confidence interval for the expected spending of motorists ...
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1answer
69 views

Linear regression involving angles in a triangle.

In a survey experiment, three independent measurements $29.5^{\circ}$, $30.5^{\circ}$, $120.5^{\circ}$ are obtained from the three angles $\alpha,\beta,\gamma$ of a triangle. Formulate the ...
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38 views

Show that: $ X \ {\text{is}}\ \mu{\text{-integrable}} \implies \sum_{k=1}^\infty\mu(\{\mid X\mid ≥ k\}) < \infty$

Assignment: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $X: \Omega \rightarrow \bar{\mathbb{R}}$ a $\mathfrak{A}$-$\bar{\mathfrak{B}}$-measurable function. Show that: $$ X \ ...
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1answer
35 views

What does the expectation value of $x$ mean? Surely it must expectation value of a function of $x$?

How can a value of $x$ have an expectation value? Surely there must be a distribution of values of $x$ for the expectation value to be calculated. Is this the reason for the normal distribution ...
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1answer
16 views

probability df of Y and expected value of Y

Let $Y =2(X-1)^2 - 1$, where $X$ is uniformly distributed over the interval $[0,2]$. Determine the pdf of $Y$ and the expected value of $Y$. so the pdf is like $f_Y(x)=\frac{f_Y}{2}$? right? and so i ...
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1answer
46 views

Convergence of Remainder from Taylor Expansion

For a distribution function $F$ and its variance functional $T(F)$, it can be shown that the Taylor expansion of $T(F)$ at $F$ in the direction of the empirical distribution function $F_n$ gives the ...
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1answer
60 views

Maximum likelihood estimator for general multinomial

Let $(X_1,\ldots,X_r)\sim\text{multinomial}(n,(p_1,\ldots,p_r))$, where $p_r=1-p_1-\cdots-p_{r-1}$. The random likelihood is $Ap_1^{X_1}\ldots p_r^{X_r}$, for some non-zero $A$. The random ...
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1answer
32 views

Let $Y_{1},Y_{2},…,Y_{n}$ be a normal distribution where $\mu =2$ and $\sigma = 4$. Find $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$

Let $Y_{1},Y_{2},...,Y_{n}$ be a random sample from a normal distribution where the mean is $2$ and the variance is $4$. How large must $n$ be in order that $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$. ...
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1answer
57 views

Lebesgue-Stieltjes Integral (Several Variables)

Let $\mathcal F$ be a convex set of probability measures or distribution functions and $F, G$ be two elements in $\mathcal F$. Let $T$ be a functional on $\mathcal F$ defined as follows. Note that $h$ ...
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2answers
158 views

Proof of the law of large numbers for higher moments

Let us work on some probability space $<\Omega,\mathscr{A},\mathbb{P}>$: I'm looking for (independent) proofs of two proofs, of the generalised weak and strong law of large numbers ...
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1answer
49 views

probability using exponential distribution

The time a student spends in the shower is an exponential random variable with a mean of 262 seconds. Calculate the probability that a student spends an average of over 270 seconds in the shower per ...
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1answer
37 views

Estimating number of slaves imported before 1790

I have a statistics problem I am having a hard time figuring out how to model mathematically. The 1790 US Census counted 697,681 slaves and 59,196 free Africans in the United States. (A) assume ...
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1answer
55 views

CLT for bounded and dependent sequence

Let $\displaystyle X_1,X_2,...X_n$ be identically distributed such that $\displaystyle Pr\{a \leq X_i\leq b\}=1$ for bounded constants $\displaystyle a,b$. Further Let $\displaystyle ...
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1answer
36 views

MLE of a random sample Y1,…,YN from a bin(n,p) distribution

I have a random sample $Y_1,..., Y_N$ from a $bin(n,p)$ distribution and I'm supposed to show that the MLE is: $\hat{p}$=$\sum_{i=1}^N Y_i \over Nn$. But when I take the MLE of the binomial ...
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1answer
45 views

Checking if pseudorandom numbers fit a normal distribution.

I don't have much background in statistics, and one of the exercises in my programming course asks for the following: Generate a sample of normally distributed data using rejection sampling and ...
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1answer
105 views

conditional probability that 5 red balls were placed in the bowl at random

Place five similar balls (each either red or blue) in a bowl at random as follows: A coin is flipped 5 independent times ad a red ball is placed in the bowl for each head and a blue ball for each ...
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1answer
45 views

PDF of uniform distribution

Find the uniform distribution of the continuous type that has he same mean and the same variance as those pf a chi--square distribution with 8 degrees of freedom. My solution: for the mean- $\frac ...