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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estimate population size from samples

I don't think there is enough data to solve for this but wanted to run it by some folks smarter than me... So given the data below where classes are a sample of a professor's entire career, can you ...
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11 views

find the the deviation score of unknown score

I'm being asked to find a deviation score for the fifteenth score when only $14$ scores are given (given are deviations from the mean which are $+16, 0, -16, +1, +2, +3, +4, +5, +6, +7, +8, +9, +9$, ...
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56 views

Hottest Days of The Year

Recently, there has been much talk in the media of it being the hottest day of the year so far. It has always seemed to me that there are likely many more of these in the northern hemisphere than the ...
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1answer
24 views

Expected value of a sample

I am unsure of how to solve this question. I know from examples questions that expected value of a sample is usually very close to the population mean. However, it says to compute the expected value ...
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1answer
22 views

Variance of subset vs total variance

Is it true that the variance of subset is smaller than variance of the total set? Given each element in the set is a N-dimensional vector, and the distance is defined as Euclidean distance. Variance ...
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1answer
46 views

German Tank Problem, Confidence Level

Suppose you're in a city with n cabs. Each cab has a distinctive number from 0 to n. You take a cab 10 times, the choice of the cabs is independent and equiprobable. The cab with the biggest number ...
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counting , please help very urgent [on hold]

There are 7 teams (A, B, C, D, E, F, G) participating in a tournament. each team has 4 members. In how many ways can the 28 members be sitted in a row such that each of the members in team A are ...
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23 views

This random variable converges in distribution?

$\delta_x$ is a Borel probability that $\delta_x(x)=1$ and $\mu_n$ is a uniform distribution in this interval $(1, 1 + \frac{1}{n})$ The variable $X_n\sim \frac{1}{2}\delta_{\frac{1}{n}} + ...
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1answer
23 views

identifying sudden change in value given a list of values over time

I have a list of the average price of an item in a game over time. Things don't tend to move much. I am wondering how I can detect whether a new value inserted is a surprising movement in price. I ...
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7 views

Check trend for non-normal data?

I have data which is non-normally distributed time series. I want to find whether with time its increasing or decreasing. How will I check it. Earlier I thought of using t-test and linear fit. But I ...
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1answer
16 views

Significance of dummy variables in probit regression

I am an undergraduate student working on some projects using probit regression. I have a question on dummy variables that I was hoping someone could help me with (which I think stems from an ...
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12 views

Comparing Two Ways of Scoring Data

I have used two different methods to give a score to the same data set. One is a discrete method and the other continuous. How do I show that the continuous method is more sensitive to changes in ...
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Establishing consistency

I need to establish the (weak) consistency of an estimator of the mean, $T=a+b\bar{X}$. I tried to apply Chebyshev's inequality, but I couldn't do much because the parameter that subtract in the ...
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1answer
16 views

Statistics probability [on hold]

A coffee shop sells 4 sizes with 4 different varieties. Customers can choose to add one or more syrups that come in 4 flavors. How many different coffees drinks can be made?
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15 views

Statistics question help! [on hold]

An insurance company classifies people to 3 groups: A, B, C. Also, according to company, the probability each group will be in an accident within a year of issuance is 0.05, 0.15, 0.3. If one person ...
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1answer
18 views

how to find the cumulative density function

Consider $$f(x)=3x^{-4} \qquad \mbox{on} \qquad x\geq 1.$$ Let $X$ be a continuous random variable on $x\geq 1$. Find the cumulative distribution $F(x)$ for $X$. I know that CDF for a continuous ...
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7answers
5k views

If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, ...
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1answer
25 views

Statistics Probability

Study shows 10% of household appliances fail during any given year. If a person has 4 household appliances what is the probability that exactly 2 of them will fail next year?
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21 views

Recursive Variance

What will be the distribution or features about the following $x$? $x=\mu+\epsilon$ where $\epsilon\sim N(0,x^{-1})$. It seems interesting in econometrics if we allow $x$ being a time series and ...
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0answers
15 views

Averaging averages and standard deviations

I am comparing exam scores for different subjects: physics, biology, english. I am looking at what proportion of the curriculum is spent on each subject. The number of students is fixed: the same for ...
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0answers
37 views

Proving linear algebra equation [on hold]

I am having trouble proving that two multivariate formulas are equivalent. I implemented them in MATLAB and numerically they appear to be equivalent. (This problem comes from Bayesian estimation, ...
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3answers
39 views

You roll 3 six-sided dice. What is the probability that the third is at least as high as the highest of the previous two?

I know that the probability of the first two dice being different is $\frac56$, and the first/second being greater is $\frac56$, but am not sure how to calculate the prob of the 3rd being greatest. ...
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19 views

business math question

Diane Van Os decided to buy a new car since her credit union was offering such low interest rates. She borrowed $33,600 at 4.25% on December 16 2012, and paid it off March 17 2014. How much did she ...
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19 views

Likelihood of large difference in averages

Note that I updated question because initial example grades did not satisfy the overall average that followed from the numbers given in the article. A newspaper artticle in Norway made me curious ...
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0answers
27 views

Interpolation at “extreme values.”

I am working with a meteorologist on a project. We are pulling data from METAR Observation stations on several variables (such as temperature, dew point, wind speed, etc.) throughout time. ...
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0answers
18 views

How to measure the stability of datas

The background: I have a server handling $n$ kinds of requests, denoted by $k_1, ..., k_n$, at a certain time, many requests has been processed, the average time it takes to process $k_i$ is $t_i$, ...
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2answers
32 views

Tabulate the probability distribution of $x$.

If a red dice and a green dice are rolled together and $X$ is the highest score minus the lowest score of the dice, what are the possible values of $X$? Tabulate the probability distribution ...
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3answers
62 views

Sample space: What's the possibility that a family has n boys?

What's the sample space in these two cases? Case1: Of all families with two children you ask the parents, if they have a boy born on a Thursday. They say yes. What's the possibility that the family ...
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1answer
13 views

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, ...
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0answers
51 views

Impact of random numbers on the eigen-values

How do the eigen-values of the following tridiagonal matrix ($A$) change when adding random numbers $R_i$ (with the condition $|R_i| < m$) to its diagonal. A is a square matrix defined as follows: ...
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1answer
25 views

Law of Iterated Expectation with Probability?

I'm trying to follow a proof of the following proposition (source) Let X and Y be two independent random variables and denote by $F_X(x)$ and $F_Y(y)$ their distribution functions. Let $$Z=X+Y$$ ...
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1answer
21 views

What to call & how to compute errors in a very asymmetric sample

Consider the following sample $\{1.25,1.5,1.75,2.0,2.25,2.5,2.75,10.\}$. Mean is $\mu=3.$, standard deviation is $\sigma\approx 2.69$. I am wondering how to compute and what to call error bars in the ...
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17 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
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1answer
23 views

Can any function of the second moment of a random variable be recovered from its quantile function?

Summary of question It is known that the expected value of a random variable can be obtained from integrating its survival function. This is easily restated in terms of the quantile function as: $$ ...
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1answer
40 views

I don't understand how I can calculate this

"Let $X_1, X_2, ...$ independent random variables. $X_n\sim B(p_n)$ and $p_n = \frac{1}{n}$. Calculate $P(lim sup (X_n=0, X_ {n+1}=1,X_ {n+2}=0 ))$" I suppose that i can use the lemma of ...
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1answer
28 views

Formula to represent 'equality'

I am trying to find the appropriate formula in order to calculate and represent the 'equality' between a set of values. Let me explain with an example: Imagine 3 people speaking on an a TV Show that ...
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3answers
63 views

Why do we calculate the median the way that we do? [on hold]

If we have 4 Apples: A A A A Then in any stats class, the medium is 2.5. I am very sure that this is wrong, and that the medium is actually 2, even though experts in the field would say otherwise. ...
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1answer
31 views

Underdetermined vs Overdetermined Problem

I'm trying to create a model which is of the form $$y = (a_0 + a_1l)[b_0+\sum_{m=1}^M b_m\cos(mx-\alpha_m)] [c_0 +\sum_{n=1}^N c_n\cos(nz-\beta_n)]$$ In the above system, $l$,$x$ and $z$ are ...
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1answer
30 views

What's the relationship between half of a value x, and the midpoint of that value?

Say you have 4 apples: A A A A Where is the middle of this, I mean, on paper it's obvious, but what is the numerical value of this? Halving this value (or rather any value) is easy, 4/2 = 2, but is ...
3
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1answer
36 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of several simplicies, with all coordinates being non-negative. That is, given $$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq \vec{0},$$ I want to ...
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0answers
35 views

Mean-Square Ergodicity of Certain Quantities?

I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way. Is it possible ...
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1answer
15 views

chi-square test as likelihood-ratio test

$X_1, \ldots, X_n$ iid $\sim N(0, \sigma^2)$. I want to deduce the chi-square test $H_0 : \sigma^2 = \sigma_0^2, ~ H_1 : \sigma^2 \neq \sigma_0^2$: Decline $H_0$, if $ \displaystyle \frac{(n-1) ...
2
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3answers
97 views

Simplest way to integrate this expression : $\int_{-\infty}^{+\infty} e^{-x^2/2} dx$ [duplicate]

I'm toying around with statistics and calculus for a project of mine and I'm trying to find the simplest/fastest way to integrate this formula : $$\int_{-\infty}^{+\infty} e^{-x^2/2} dx$$ I do not ...
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1answer
20 views

finding a question about constrained regression - had a side constraint $x \geq y^*z$

I saw a question that asked about solving a non-negative least squares problem with $3$ unknowns, $(x,y,z)$. But there was an additional constraint, $x \geq y^*z$. Would appreciate getting the ...
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0answers
23 views

removing a factor when the summation equals zero

http://www2.warwick.ac.uk/fac/soc/economics/staff/vetroeger/teaching/po906_week567.pdf On page 9 of the above link. I don't understand the step from row 5 to row 6. After the x_i is pulled out as a ...
2
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0answers
63 views

Why is this the solution?

I have this exercise with the solution. Let $X\sim N(0,1)$. Show that $P(2X = 3Y + 1) = 0$ if $Y\sim \text{Poisson}(\lambda)$. I have this solution $P(2X=3Y+1)= P(\bigcup_{k=0}^{\infty}(2X = 3Y+1, ...
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0answers
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Linear Combination of the Normal Distribution Two missing variables? [on hold]

An insurance office buys paper by the ream, 500 sheets, for use in the copier, fax, and printer. Each ream lasts an average of 4 days, with standard deviation 1 day. The distribution is normal, ...
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1answer
20 views

What is meant by statisticians when they talk about between population differences vs within population differences?

Suppose we have two populations of people in different parts of the world and we want to talk about the variation in heights between the two populations. As I understand it, statisticians are talking ...
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0answers
21 views

Median of the mean/max of an iid sample of exponential variables

I would like to know if one can obtain a simple analytic expression of the median of $T_n$ defined by $$ T_n = \frac{\overline X_n}{X_{(n)}}, $$ where $\overline X_n$ is the empirical mean and ...
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1answer
31 views

Expected value of discrete functions. [on hold]

I am doing some research in number theory(High school-so nothing advanced). During this I came across this post. I have not done much statistics. So could someone explain to me why if $\displaystyle ...