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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Problems in Quartiles.

above question is copied from http://web.mnstate.edu/peil/MDEV102/U4/S36/S363.html http://www.alcula.com/calculators/statistics/quartiles/ I have calculated Quartiles using above method (method ...
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3k views

Likelihood Ratio Test for Linear Regression

I apologize for the image I am posting below. I am new to StackExchange and I am not yet familiar with the MathJaX equations, so I took a screenshot. Here is my question: Let the independent random ...
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27 views

Expected value of maximum likelihood estimator of a Bernoulli random variable

While reading the text from Keith H. Thompson on the Estimation of the Proportion of Vectors in a Natural Population of Insects, I came across the following part where I don't understand everything. ...
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31 views

When one can cancel out integration from both sides?

For example, can we go from $$ \int_0^\infty e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )f(ε_i )dε_i=δ\int_0^\infty f(ε_i )dε_i $$ to $$ e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )=δ $$ by cancelling out integration from ...
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How to solve an integratation involved an unknown function?

Can anyone have any suggestions how to solve this equation for $w_i$, that is, what is the solution of $w_i$? $$ \int_0^\infty e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )f(ε_i )dε_i=δ $$ Where, $f(ε_i)$ is the ...
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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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21 views

Finding Variance using MGF with sums of R.V's

Consider a r.v. X with MGF $m(t) = \frac{e^{k(e^t-1)}}{(1-bt)^a} $ $(a,b,k) ∈ \Bbb{R}$ Differentiating twice we retrieve $m''(t) = ...
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44 views

How do I calculate a combined data error?

I have a time series of data points read from an instrument, I also get an estimate of the error ($\sigma$) for each of these readings. e.g. $10.12 \pm 0.05$ $10.09 \pm 0.02$ $10.20 \pm 0.12$ ...
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400 views

How do I calculate a point estimate of the largest 10%?

Here is the provided data: The question asked is: Calculate a point estimate of the value that separates the largest 10% of all values in the thickness distribution from the remaining 90% and state ...
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461 views

What is the probability that a randomly selected 18-ounce bag of chocolate chip cookies contains fewer than 1300 chocolate chips?

I have no idea where to start on this question or where to begin to find the answer. And I'm not sure how to calculate the probability. Here is the full question: The number of chocolate chips in ...
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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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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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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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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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539 views

Show that -log transformation of Pareto distribution is exponentially distributed.

Question: Given that y is distributed as: $$ f(y; \theta) = \theta y^{(\theta-1)} $$ $$0<y<1 , \theta>0$$ If Z = -log(Y), show that Z has an exponential distribution.(ie $E(Z) = ...
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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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737 views

How did Target figure out a teen girl was pregnant before her father did?

First of all I do not have a mathematics degree only a B.S. in finance so please take that into account when writing an answer. Generally what type of mathematics is involved here? And specifically ...
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480 views

Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
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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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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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38 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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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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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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4k views

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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22 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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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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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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40 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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33 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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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
398 views

Measuring degrees of randomness

Imagine, for simplicity's sake, that we have a set of numbers, each equal to either 0 or 1. Let's call each a bit. Rationally, if the set is completely random, and reasonably large, the probability ...
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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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518 views

How do I compare student pre-test scores with post-test scores to evaluate whether or not they “learned”?

We need to track student advancement in a topic based on pre and post test scores. That is, we give a pre-test on day 1 of class, then on the last day we give the exact same test, renamed as a ...
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1answer
1k views

How to calculate lower & upper quartiles?

I'm sure this has been asked many times before but it's confusing me a lot so hopefully someone can help! I am given this data set: 0.28 0.30 0.42 0.59 0.71 1.67 1.82 2.39 4.71 4.79 4.89 5.00 5.00 ...
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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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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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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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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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621 views

Covariance- v. correlation-matrix based PCA

In principal component analysis (PCA), one can choose either the covariance matrix or the correlation matrix to find the components. These give different results because, I suspect, the eigenvectors ...
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1answer
41 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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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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93 views

Suppose that $E[X^n] = 3n$. Find $E[e^X]$…

Suppose that $E[X^n] = 3n$. Find $E[e^X]$. Hint from my professor: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +···$ Not quite sure how to solve this problem, wouldn't $e^x$ go on exponentially. ...
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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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96 views

Optimal strategy in a simple card game

So some friends and I are playing this simple card game. We choose 15 random cards from a normal deck of card and lay them on a table in a pyramid shape. 5 cards in the bottom row, 4 in the next row ...
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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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1answer
15 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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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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10 views

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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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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69 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: $\frac{{7\choose2} 5! 6*1}{6^7}$ ...