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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9 views

Show that $S^2$ is an unbiased estimator.

I haven't worked with unbiased estimators much, where do I start with this? With sample variance defined as $S^2 = (\sum_{i=1}^n (X_i - \overline{X})^2 )/(n-1)$ A. Show that $E(X_i^2) = \sigma^2 + ...
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4answers
35 views

Minimizing a function - sum of squares

I'm hoping you can help with this problem. I haven't taken calculus in years and I don't know where to start... The sum of squares of a sample of data is minimized when the sample mean is used as the ...
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1answer
13 views

Finding probabilities from probabilty generating function

Given that I have a probability generating function for $Q$ given by $\dfrac{4s^{2}}{9-3s-2s^{2}}$, I want to find $P(Q = n)$ for $n \geq 2$. I understand that I could actually use the definition of ...
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6 views

testing independent samples

I have 2 samples in statistics. I have to choose one of the two possible options paired or two sample t-test. What must I check? I think, paired test means that both must have normal distribution, ...
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9 views

The uniqueness of solution of an equation that involves CDFs

I have two monotone CDFs $F(x)$ and $G(x)$. The functions are symmetric in a sense that $F(x)=1-G(1-x)$, $f(x)=g(1-x)$. I am trying to show that equation $xF(2x)+(1-x)G(2x)=1/n$, $n\geq2$ has a unique ...
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1answer
25 views

Determine relation of $x$ and $y$ from results

I can't seem to determine the relation between $x$ and $y$ for this problem. All of the previous ones I have done have been doable simply by eye-balling the relation between $x$ and $y$, but here I am ...
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1answer
60 views

Can you simplify this?

From there : $$\Large{ \int_{-\infty}^{+\infty} \frac{1}{\sqrt{2\pi} \sigma_x } e^{ -\frac{(x-\mu_x)^2}{2\sigma^2_x}} \frac{1}{\sqrt{2\pi} \sigma_y } e^{ ...
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28 views

Positive definite function and covariance matrix.

I tend to view positive definite function as a function of elements of positive definite matrix. A reference is: https://en.wikipedia.org/wiki/Positive-definite_function My question in essence: is ...
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1answer
30 views

Distribution of minimum absolute value

Consider $K$ independent Laplace variables $X_k, k=1,\ldots,K$, with mean 0 and scale $\lambda$ (so that their PDF is $f(x)=\frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}}$. Let $Y$ be the variable taking ...
2
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1answer
31 views

Probability theory required for learning statistics rigorously

I would like to learn statistics rigorously. The only book that I can find that seems to do statistics rigorously is this book "Theory of statistics" by Schervish (which seems advanced): ...
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2answers
39 views

Determining probability generating function for waiting time to see first $SS$

Given a sequence of Bernouilli trials, we have $P(S) = \frac{2}{3}$ with $0<p<1$. The event "SS" occurs on the $i$-th trial if we observe an $S$ on the $i$-th trial following a $S$ on the ...
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1answer
11 views

Stem and leaf plot Standard deviation question

This is a question cropped out of a midterm practice exam. It states to calculate the standard deviation but I'm confused because it would take me a significantly long time calculating this under the ...
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0answers
17 views

Don't understand formula to use

The Citiburg Parks Department has just installed 1400 new lightbulbs with an expected mean lifespan of 75 months and a lifespan standard deviation of 6 months. How many bulbs will need to be replaced ...
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4answers
57 views

Probability question involving infinite number of vertical chords in a 1 inch circle. [on hold]

Infinite number of vertical chords drawn on a circle with a 1 inch radius. What is the probability that a randomly picked chord is shorter than the radius? The answer should be $1 - .5√ 3$ or ...
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1answer
43 views

An isosceles right triangle has legs of length 10. A pin is dropped into it and lands somewhere in the triangle where all places are equally likely.

What is the probability that it does not land within 2 units of any of the sides? From my calculations, I get that the smaller triangle has side lengths of 4,4, 4 root 2 (-2 at the right angle and ...
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2answers
38 views

A grasshopper starts at the origin and is equally likely to hop north,s,e,w. What is the probability that it's coordinates will be 0,0 after 4 hops?

The grasshopper must hop in all $4$ directions (North, South, East, and West) to get back to the origin after $4$ hops. Therefore, I did: $\frac{(4 \cdot 3 \cdot 2 \cdot1)}{4^4} = .09375$. However, ...
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0answers
10 views

Probabilistic Modelling of uncertain positions of objects in a 2D-Grid

I have a 2D-Grid which is populated by obstacles of different sizes. A size is always a whole number of cells. An obstacle is at least one cell big. If I did kown the size of the object but had only ...
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0answers
16 views

GARCH model, expectation of volatility?

Consider a time series $\{r_t\}$ following a standard GARCH(1,1) model, i.e., $$ r_t = \sigma_t \epsilon_t,$$ where $\epsilon_t \sim N(0,1)$ and are i.i.d, and $$\sigma_t^2 = \omega + \alpha_1 ...
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0answers
10 views

On statistical analysis and sudden changes in data

Here we see the value of Euro against the United States Dollar, provided by BBC approximately 10.00 GMT on the 6th of June, 2015. On the 5th Greece had a referendum, and it's outcome of "No" to ...
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1answer
31 views

What are conditions under which convergence in quadratic mean implies convergence in almost sure sense?

What are the conditions on the sequence on $\{X_n\}$ (apart from the degenerate random variable), under which it can be claim that $||X_n-X||_{L^2(\mathbb{R})}\rightarrow 0$ implies $X_n\rightarrow ...
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1answer
28 views

How is randomness quantified in Bayesian Statistics?

How is randomness quantified in Bayesian Statistics? In the finite case of N items, it is simple, since I can assign a probability of 1/N to each of the item. However, I wonder what happens if I want ...
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0answers
12 views

hypothesis testing ; F-testing

lnQ=1.37+0.632lnK+0.452lnL (0.257). (0.219) cov(bk,bl)=0.055 R^2=0.98 H0: bk+bl=1 F=(Rb(hat)-r)'[R(X'X)^-1R']^-1(Rb(hat)-r)/e'e/n-k-1 I found the numerator value but i ...
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0answers
17 views

Calculating probability of an event using mean and stnd deviation [on hold]

A factory produces electrical devices. It is known from experience, that the life-time of this devices will be normal distributed. With:  Mean = 1100 hours  standard deviation = 50 hours. ...
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1answer
26 views

Proof of equivalent probabilities in anomaly detection

In A New Look at Anomaly Detection there is a claim for the proof of probabilistic definition of normal is as follows, a guess of the probability for event i is $\pi_i$, the true probability is ...
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2answers
17 views

When do I use a z-score vs a t-score for confidence intervals?

I have a set of 1000 data points. I would like to estimate their mean using a confidence interval. I read somewhere that if the sample size, $n$, is bigger than 30 you should use a t-score, and else ...
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3answers
29 views

For non-negative data the sample mean is not smaller than its standard error.

(1) Let $X_1, X_2, \dots, X_n$ be a random sample from a population with non-negative values. Then show that $\bar X \ge S/\sqrt{n},$ where $S^2 = [\sum_{i=1}^n (X_i - \bar X)^2]/(n-1).$ I have not ...
29
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3answers
690 views

Guessing the length of a playlist on “shuffle random?”

The other night I was hanging out with some friends and someone put on a playlist on shuffle random, where the songs are drawn uniformly at random from a fixed playlist. The person who put the ...
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1answer
30 views

Statistics - Exponential distribution

There are $n$ machines. Each has durability given by exponential distribution with $EX = 10$. If a dead machine is replaced with new one immediately, find minimal $n$ so we can say with $P = 0.99$ ...
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7 views

Moving average where periods have unequal # of samples

I'm trying to compare a simple moving average approach to one that normalizes by the number of samples in a period to determine which is "more correct." Here's a representative piece of the data: ...
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1answer
23 views

Computing the Density of Points on a 2D Plane

I have a chart with points scattered across it; many of which are tightly clustered. I wish to filter out all points that are not within these dense clusters of points. I have an idea of how to ...
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1answer
23 views

Difference between stochastic process and chaotic system [on hold]

Can anyone please point out some difference and similarity between stochastic system and chaotic system?
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1answer
18 views

How do you manage fractional counts in a binomial calculation?

How do you choose the counts to include in a binomial calculation if the range of counts is not discrete? For instance, given a binomial distribution of 110 counts and a probability of 0.301 of ...
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1answer
40 views

Statistics - normal distribution problem

Two random variables $X$ and $Y$ are i.i.d. normal$(\mu, \sigma^2)$. If $P(X > 3) = 0.8413$ find $P((X+Y)/2 > 3)$. The result must be exact number, so normal distribution parameters are ...
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21 views

Probability of selecting a sequence in order

If "X" number of attempts are made by "Z" number of persons to select a random number from a range "r", where "X <= r". Then I am interested in finding the probability that a particular sequence in ...
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1answer
28 views

Is the mode of the sum the same as the sum of the modes

If we have $N$ sets, $\{A_{1},\dots, A_{N}\}$, and we form a set $S$ by taking the sum of each element in the set with each element in the other sets, what can we say about the mode of $S$? ...
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1answer
65 views

Why does this expectation integrate to 1

Let $p(y|\theta )$ be our likelihood, and $\hat{p}_{N}(y|\theta)$ be an unbiased estimator of our likelihood. Let $z=\ln \hat{p}_{N}(y|\theta) - \ln p(y|\theta )$, and $g_{N}(z|\theta)$ be the ...
2
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1answer
33 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
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1answer
21 views

Two types of errors, type-$1$ error and type-$2$ error, can not be minimized simultaneously when the sample size $n$ is already fixed. How?

I read in some of the books that the two types of errors, type-$1$ error and type-$2$ error, can not be minimized simultaneously in Neyman Pearson Theory of testing of hypothesis when the sample size ...
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2answers
40 views

Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?

Here's what I have so far: All 4 same on first try = (1/4)^4 * 4 3 same, then get 4th on 2nd roll = 4 * (1/4)^3 * (3/4) * (4!/3!) Here's where I'm confused: 2 same = 4 * (1/4)^2 * (3/4)(2/4 :to ...
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1answer
23 views

How to find which treatment is most effective in gene data given one standard method and 3 variations

Sorry I am a biologist and it appears am not quite confident enough for statistical analysis. I have datasets that represent different treatments on a biological system. It records how many genes have ...
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1answer
29 views

Confidence interval of a uniform distribution

I need some help with the following problem: I want to estimate $n$ of $X_i \sim U(1, n)$ with a $90\%$ confidence level. What is given is the sample size with $10$ and the maximum of the sample with ...
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2answers
42 views

12 six-sided dice are thrown. What is the probability of getting each number twice?

I got this: $\frac{6!12!}{6^{12}2!^6}$ but the answer is this: $\frac{{12!}}{6^{12}2!^6}$ Im not sure I understand why you wouldn't write the $6!$ because if the first die's value is #3 then you have ...
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1answer
18 views

probability of a proportion point estimate

I've got a problem where I'm supposed to find the probability of a point estimate but cannot see how my answer is differing from the given one. The problem is: Unknown to an experimenter, the ...
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1answer
32 views

On a 50 question multiple choice exam with 5 choices per questions, What are the odds that I get 100% if I were to Guess every answer? [on hold]

What would the odds be to get 100% on a multiple choice exam where I guessed the answer to all 50 of the multiple choice questions (5 choices per questions)? A 1 in how many chance?
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0answers
16 views

LRT and Wald Test in Multivariate Linear Regression significance

I've been researching all afternoon trying to get a better idea of what the Likelihood ratio test and the Wald test are actually doing. I have a bunch of covariates and I'm testing out like 30 ...
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0answers
23 views

A Question about the Kurtosis

Problem: Show that if a binomial distribution with $n = 100$ is symmetric, its coefficient of kurtosis is 2.9. Answer: First, I am interpreting the term symmetric to mean that $p = q = \frac{1}{2}$. ...
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1answer
44 views

Question with excel [on hold]

Suppose I have a set of number like: 1,2,4,3,4,2,5,3,7,1,2,2 1) How can I make an histogramme with proportion on excel ? i.e. there is 4 number 2, 2 number 1... 2) add over the curve of the normal ...
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1answer
30 views

Geometric sum of geometric random variables

I am looking to find the probability mass function of $Y=\sum_{i=1}^NX_i$ where $X_i\sim\textrm{Geometric}(a)$ and $N\sim\textrm{Geometric}(b)$. I attempted to do this by finding the probability ...
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1answer
25 views

Hypothesis Testing? Probability

I'd be really grateful if someone can help me. I found this problem in the textbook Probability and Statistics by Wapole. This question look something similar to a Hypothesis test, but I have no ...
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0answers
25 views

Closeness in distribution implies closeness in statistics?

I am aware that convergence in distribution does not necessarily imply convergence in the mean. I browsed through some examples of statistical distances here ...