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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How to Find the maximum likelihood estimates of the parameters for a normal distribution with expectation 0 and covariance matrix [closed]

Is there anybody to help me on one exercise in Mathematical statistics? it is related with bivariate normal distribution ,
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22 views

The conditional probability density function with a specific condition [on hold]

Assume the following discrete time model: $x(t+1)=Ax(t)+w(t)$ where $w(t)$ is zero mean, iid white noise with bounded covariance matrix $Q$. Let $s=x(t)+x(t-1)+x(t-2)$. How I can find ...
3
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1answer
29 views

Prove that if $X_2$ has a uniform distribution then $X_1 \oplus_2 X_2$ too

Assume we have two independent random variables $X_1$, $X_2$ with values in the set $Z_2 = {0,1}$. Prove that if $X_2$ has a uniform distribution then $X_1 \oplus_2 X_2$ has also the uniform ...
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0answers
12 views

OpenBUGS: get a sample from a random variable

I'm working in OpenBugs, and I've defined the next model: $Y\sim {\rm Exp}(\theta)$ so I'm asked to assign different initial distributions to $\theta$: Normal, Gamma and log-normal, to this point ...
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33 views

How it is shown by the following integral?

Example: Ornstein-Uhlenbeck Process. Let $ dx=-\eta xdt+\sigma dz $ be an Ornstein-Uhlenbeck Process Write the moment-generating function for $x(t)$ as $$ M(θ,t)≡E(e^{-θx})=∫_\infty^∞ ...
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0answers
10 views

How to measure number values over time for behavior change

I am not a math expert at all and looking for some advice. I have a data set which has a time and number value for 5 minute intervals. I'm looking to detect behavior changes in the number patterns ...
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0answers
28 views

How to find expectation of Binomial Mass Function?

For example, $$ E \scriptstyle\binom{n}{r}\Phi(X)^r(1-\Phi(X))^{n-r} $$ Where X follows normal distribution with mean $\mu $ and standard deviation 1, and $\Phi(.)$ is the normal CDF. Thank you
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1answer
25 views

Calculating the Confidence Interval

It is found that in a random sample of 100 Science students, there are 48 studying statistics . To test whether the true proportion of students in statistics is 50% or not, suitable null and ...
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5 views

Expressing a trial-and-error answer algebracally

I have a question where it's about statistics. (x) 1 2 3 4 5 (f) 5 10 $p$ 6 2 The median is 3 and the mode is 2. I need to find 2 possible values for $p$ If mode is 2 then $p < 10$ and by ...
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51 views

Probability Question Statistics [closed]

An urn contains seven black balls and three white ones. Two players, A and B, take turns to draw a ball at random without replacement, until one wins by drawing a white ball. Player A draws first. ...
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35 views

decision making [closed]

Assume that the average admission for all hospitals in Melbourne is 7500. Conduct a statistical hypothesis test to determine if the admission of hospitals in Melbourne is significantly different from ...
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1answer
27 views

Proof that there will always be data within 1 SD

So we just started stats at school, and our teacher told us that no matter the data, no matter how distorted or weird it is, there will always be data within 1 standard deviation of the mean. Is this ...
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0answers
36 views

Finding for X in a Probability Equation

I've come up with an equation for my job. Currently it solves for z, the probability. I need it to solve for X, given the probability. This will be used in a computer program. The equation is as ...
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0answers
20 views

What would be the standard errors of this transformed regression model, given that I know the standard errors of the original model

Say I have the following regression model: $\ln\left(\dfrac{y_i}{x_{2i}}\right)=\alpha_1+\alpha_2\ln(x_{2i}) + \alpha_3\ln(x_{3i}) +e_i$ where I know the values of the regression coefficients and ...
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3answers
42 views

Total Law of Probability Question

In a community $25\%$ of the residents are smokers. Suppose $30\%$ of the smokers claim that they don’t smoke, and all non-smokers say they don’t smoke. What is the probability that when someone says ...
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24 views

Linear or nonlinear modell

Given those three modells and the assignment to decide whether or not those modells can be transformed into linear modells: (a) $Y_i = \beta_0 + ...
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1answer
14 views

Alternative formula for sample covariance [closed]

Is this an equivalent formula for the sample covariance? $\frac{1}{n-1}(\sum_{i=1}^nx_iy_i -n\overline{x}\overline{y})$ Thanks
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1answer
20 views

Risk function for vectors

How do apply risk functions to vectors? Here is the problem I have encountered: Let $X = (X_1, X_2, . . . , X_p)$ be a collection of independent random variables with $X_i \sim N(\mu_i, 1)$ for $i = ...
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0answers
5 views

Is yt is weakly stationary?

I have a model $y_t =0.5y_{t-1} + x_t +v_{1t}$ where $x_t=0.5 x_{t-1} + v_{2t}$ and $v_{1t}$ and $v_{2t}$ follow IID normal distribution ∼ $(0,1)$. I need to show if $y_t$ is weakly stationary or not. ...
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0answers
24 views

How to test a weatherman's predictions?

I was thinking about how one may go about statistically testing a weatherman's predictions for accuracy. It's an interesting problem, because a weatherman will (almost) always deal in probabilities, ...
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39 views

Confidence interval problem!

One day the service center received $125$ calls. Out of these calls a random sample of $40$ were drawn with the average call length $7.28$ mins and the standard deviation $5.32$ mins A $90\%$ ...
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31 views

$n$ times integrated Brownian motion martingale process

According to this post, we found that a $n$ times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ...
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4answers
18 views

Find the total amount of beads used if there is twice the number of large beads than a small one?

I had a test a few days ago (year nine level) and my friends and I were stuck on this question: "Bill is making a bracelet using small and large beads. There are twice the number of large beads than ...
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1answer
36 views

Solving a numeric statistics problem - R [closed]

I am quite stuck with solving some complicated numerical equation I would like to solve the following equation: $(1-k)\tilde{\alpha}+kf(\tilde{\alpha})=C$ where $0<k<1$ and ...
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0answers
19 views

Simple example of “Spike-and-Slab Prior” for Bayesian Inference

I would really like to understand how Spike-and-Slab Priors work in relation to Linearized Models. Can somebody provide a toy example of a Spike-and-Slab Prior with a Bernoulli spike and a Gaussian ...
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1answer
19 views

In the context of linear regression with two parameters, how do I show $\det(\mathbf{X}^\text{T}\mathbf{X} )\ne 0$?

Let $\mathbf{b} = \begin{bmatrix} b_0 \\ b_1\\ \end{bmatrix} $ and $\mathbf{X} = \begin{bmatrix} 1 & x_{11} \\ 1 & x_{21} \\ \vdots & \vdots \\ 1 & x_{n1} \end{bmatrix}$ Then ...
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0answers
35 views

Suppose $x_n$ is a sequence of positive monotonically increasing random variables converging to $X$. Show $\lim_{n \rightarrow\infty}E(x_n)=E(X)$

I am hoping to get some verification of the below proof. I am worried that I am missing something conceptually. That $\lim_{n\rightarrow \infty}E(x_n)\leq E(X)$ is clear since $E(x_n)\leq x$ for any ...
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0answers
17 views

Does increasing sample size have any effect on omitted variable bias?

Say I have a multiple linear regression model, where two of the variables are positively correlated, and I omit one of these variables from the model. First question - if I increase the sample size, ...
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2answers
22 views

What is an example of a second-order markov chain? [closed]

I'd like to see an example of a second-order markov chain. Haven't found one over google or in any of my textbooks
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1answer
12 views

Joint probability of bivariate discrete

The number of people who enter a car dealership, X, is either 1, 2 or 3 each with probability 1/3. The number of people who buy, Y, given that X people enter the dealership is binomial with n=x and ...
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18 views

Coefficient Correlation r of Exponential Functions Regression

I'm writing an exponent regression calculator $Ae^{Bx}$ Sample Data Set (X,Y) is (9, 1) (7, 10) (6,11) (20, 10) (15, 1) A = 5.287 and B = -0.0232. So $F(x) = ...
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1answer
40 views

Joint pdf of two transformed variables ($W$ and $Z$) from joint pdf of $X$ and $Y$.

Let the joint distribution of $X$ and $Y$ be given by: $f(x,y) = e^{-x}$ if $0 < y \leq x < \infty$ Define $Z = X+Y$ and $W = X-Y$ Find the joint pdf of $Z$ and $W$ Calculate $f_{ZW} ...
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Analysing error in Multiple Regression Analysis. [closed]

Hello everybody, I have the following multiple linear regression model LN(Number_of_person_in_househol)=1.514-0.13(Age_of_respondent)+0.486 ...
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0answers
17 views

Error in estimated time-delay between two histograms

Suppose two histograms are made from sampling from the same probability density function $p(t)$, but one histogram is shifted with respect to the other. Is there a lower bound one can compute (for a ...
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1answer
14 views

Central limit theorem to estimate probability that estimate is larger than 25%??

In a city, 20% of the people smoke but I don't know this value. To estimate it, you conduct a survey to 1000 people if they smoke or not. Use the central limit theorem to estimate the ...
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weakly stationary time series - what type of model is this?

Say I have the following model: $y_t = 0.5y_{t−1} +x_t +v_{1t}$, and $x_t = 0.5x_{t−1} +v_{2t}$, where both $v_{1t}$ and $v_{2t}$ follow IID normal distribution ∼ (0,1). How would I go about showing ...
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2answers
68 views

Proof: Probability of a pair when rolling 7 dice is 1.

I know that the probability should be over or exactly 1 since out 6 possible values the 7th dice will always be a duplicate. My calculations are wrong though: $\frac{{7\choose2} 5! 6*1}{6^7}$ ...
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1answer
20 views

Using the Gamma Distribution

A company employs $N$ people, with their collective annual income in pounds being a random variable with Gamma distribution with parameters $\beta = 1$ and $\alpha = 90000 N^{1/2}$. For each employee ...
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0answers
23 views

Chi-square test to check sampled variance

I have two independent unknown points $x, y \in \mathbb{R}^3$ and a set of $N$ observations $x_i$, $y_i$ of their positions that I model with a normal distribution: $x_i \sim \mathcal{N}(x, ...
2
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1answer
28 views

discretized Brownian motion

These are the definitions I'm working with: A (standard) Brownian motion in $\mathbb{R}$ is a stochastic process $W(t)$ $(t \geq 0)$ such that the following properties hold: $W(0) = 0$ almost ...
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34 views

Can I assume a population is discrete if the probability of each event happening is reasonably high?

For instance, if I am presented with data that $P(X=50)= 0.1$, $P(X=60)= 0.2$, $P(X=70)=0.5$, $P(X=80)= 0.2$ Can I assume that it is a discrete distribution, given the fact that if it is a ...
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1answer
44 views

variance of a random variable on unit interval with mean inequality

Let $X$ be a random variable restricted to the unit interval (ie for probability space $\Omega$, $X:\Omega \rightarrow [0,1]$). By the Popoviciu Inequality, I have an upper bound on the variance of ...
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19 views

Finding the CDF from a PDF with multiple conditions

The PDF is \begin{align} f_x(x) = \begin{cases} \frac{6 x^{\frac{1}{2}}}{13} & \quad \text{if } 0 \le x \le 1 \\ \frac{6x}{13} & \quad \text{if } 1 \le x \le 2 \\ 0 ...
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1answer
14 views

What is the distribution of the natural numbers in the list of the sum of their digits taken in binary representation?

I'm wondering what is the distribution of the numbers in the list of the sum of their digits in base $2$. To be clear on what I mean is that if you take the $n$ first natural numbers (without zero), ...
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15 views

Find p-value? Multiple regression analysis

I want to know can I find the approximate p-value for (beta)𝛽2 if you were to test a hypothesis? How do I do that? \begin{align} n & = 30 \\ \hat y & = 123.2 + 4.59x_1 + 1.25x_2 − 6.04x_3 \\ ...
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What are the causes of overfitting in regression/classification for statistical data?

Say I have some n-dimensional data, and I want to come up with some hypothesis function which generalizes that data for future predictions in the model. "Overfitness" of my hypothesis function is a ...
0
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1answer
18 views

Random variable X with values on [0,2]

The CDF is $c(3{x^5} - 15{x^4} + 20{x^3})$ and I'm supposed to say which constant c makes it a well defined CDF. Wouldn't any integer make it a well defined CDF? I don't understand. I'm also ...
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22 views

Sample median of sample from uniform distribution.

I'm trying to determine the distribution of the sample median of a sample of size $n$ from the uniform distribution (on the interval $(0,1)$). If $n$ is odd and $m=(n+1)/2$, then the sample median is ...
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33 views

How to create a ranking system in Statistics (like ELO rating system in chess)?

I need to know what kind of information, when it comes to the math part, is needed in order to create a rating system (like the API ranking system, or the ELO rating system in chess). When it comes to ...
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14 views

Stationary time series (mean, variance, confidence intervals)

Say I have the following model: $y_t = 0.5y_{t−1} +x_t +v_{1t}$, and $x_t = 0.5x_{t−1} +v_{2t}$, where both $v_{1t}$ and $v_{2t}$ follow IID normal distribution ∼ (0,1). 'The unconditional variance ...