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 determine if a result is random variation or an unusual event [closed]

Question is: over the period of 27 years an average of 81 new cases of cancer was diagnosed per year. For the 28th year the count was 108 new cancer patients, do appropriate calculations to determine ...
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12 views

Sub gaussian concentration for Lipschitz functions

It is well know that: if $f:\mathbb{R}^m\to\mathbb{R}$ is a Lipschitz function with Lipschitz constant $L$, and $X_1,\dots X_m$ are i.i.d random variables s.t. $X_i\sim N(0,1)$, then for any $t>0$ ...
3
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3answers
73 views

Arrangements of Chairs in a Circle

Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. Hints only please! This is a confusing worded-problem. We ...
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0answers
42 views

Bartlett's paradox in Bayesian evidence

I've come across Bartlett's "paradox" (not to be confused with Lindley's paradox, also known as the Lindley-Bartlett paradox) in Bayesian statistics. The paradox originates from Bartlett's 1957 paper, ...
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0answers
18 views

L1 and L2 regularization and L1 and L2 space

I am looking to characterize the difference of the function obtained in the solution process of $L^1$ and $L^2$ regularization. It is known that $L^1$ regularization gives sparse solutions. In $L^2$ ...
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1answer
26 views

Derive State Transition Matrix from an unscented transformation

I have an application where I am using an unscented Kalman filter to process data. While the unscented transformation eliminates the linearization assumption used with the typical state-transition ...
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1answer
25 views

How to find total numbers of people with height higher than some value?

Is a continuous distribution. The y axis is not probability, how do I find how many people with height higher than some value? Now assume it is an even distribution. y is no. of people, x is ...
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1answer
25 views

How to calculate CI for percentage difference?

Suppose I have some paired data $(X,Y)$ and I wish to check if X is different to Y. To do this I could simply use the paired t-test using the difference $(X_i - Y_i)$ for each $i$. I can then use ...
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2answers
36 views

Intuition of joint density of min(X,Y) and max(X,Y)

The problem is to find the joint density of $U = min(X,Y)$ and $V=max(X,Y)$ when both are exponential random variables. The solution to it is: I can finish it after the first step but I don't ...
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0answers
24 views

Statistical Significance

I'm attempting to compute statistical significance for two data sets. I have the mean and the number of data points, but I don't think I can compute std deviation, because I don't have the individual ...
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1answer
11 views

The value of z representing the first Quartile of the standard normal distribution is:

I'm in desperate need of a hint at how they got the answer.
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1answer
29 views

Normal Approximation to the Binomial (Multiple Choice Question)

My first instinct in this question is use Normal approximation because N is large, and P is exactly between 1 and 0. I used the normal approximation, calculated when $p(X\le 19)$ and got 0.8997. The ...
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1answer
36 views

Difference of Ordered Uniform Random Variables

Let $X_1, X_2,..., X_n$ be $n$ random variables distributed uniform(0,1) and $X_{(1)},X_{(2)},..., X_{(n)}$ be the ordered statistics of $X_1,...,X_n$ such that: $X_{(1)} < X_{(2)} < ... < ...
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1answer
92 views

Combinatoric meaning of $\binom{n}{k}$

$$\binom{n}{k}$$ Means how many ways there are to choose $k$ objects out of $n$ objects (order of picking doesnt matter). But does $\binom{n}{k}$ also mean how many ways there are to arrange ...
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3answers
62 views

Why is the probability multiplied by $\binom{n}{k}$

A while ago I asked a question about probability here Why is binomial probability used here? I get that you can find how many ways of choosing the $6$ correct out of $10$ questions. But why do we ...
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1answer
39 views

Distribution of the maximum of absolute value of multivariate Gaussian

I am currently working on some simulations. However, I encounter a statistical problem as following. Suppose $ 0 < t_1 < t_2 < \dots < t_m < 1 $ and $ B(t) $ denotes Brownian bridge. ...
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1answer
21 views

I have a statistics homework problem how to find new sample mean

A random sample of 10 bank customers is asked how frequently they make credit card purchases each month. The resulting sample mean is x=102.5. If an 11th observation of 145 credit card transactions ...
2
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1answer
75 views

Least-squares solution to a matrix equation?

Suppose I have $n$ observations of $m$ dependent variables $y_1,\dots,y_m$, and I believe they follow some model wherein they can all be written as linear combinations of some underlying variables ...
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1answer
51 views

Probabilities: $P(A)=1/2$, $P(A|B)=4/7$, $P(B|A)=2/5$, $P(A \cup B)=?$

I've literally spent the past 5 hours trying to figure this out, and I just can't understand where some of these numbers are coming from. For example- If $P(A) = 1/2, P(A|B) = 4/7$, and $P(B|A) = ...
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0answers
25 views

Mutual information as a fraction of entropy?

Suppose I have two (discrete) random variables $(X,Y)$ with some joint distribution $P$. The mutual information $I(X;Y)$ is informally defined as the reduction is the remaining entropy in $X$ once the ...
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0answers
25 views

Survival analysis with a parametric model for recurrent events and time-dependent covariates

My goal is to model waiting times between recurrent events with time-dependent covariats with parametric models (Poisson, Weibull, log-normal etc.). This is not an issue time-dependent covaraits as ...
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1answer
29 views

Understanding degrees of freedom for the chi-square component of the t-distribution

In looking at the t-distribution of $\sqrt{n}(\bar{x}-\mu)/S$, I can see that this equates to the following: $$\frac{\sqrt{n}(\bar{x}-\mu)}{S} = \frac{(\bar{x}-\mu)}{\frac{S}{\sqrt{n}}} = ...
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0answers
21 views

Normal Distribution and optimization

Suppose the radius $X$ (in mm) of certain kind of water pipes follows the normal distribution $N(\mu,1)$. If the radius is less than 10 or larger than 12, then it is failed product. Suppose the ...
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1answer
38 views

Basic Probability blood types

I just want to confirm this. $ p(A \cap$ B) is not mutually exclusive since the event AB is able to occur. However $P(O\cap B)$ can not occur because there is no event OB. Here is how I am thinking ...
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1answer
17 views

Normal approximation to Binomial probability distribution

Where did this 0.5 come from? I understand we are using Z-score but in my calculations I basically omit the 0.5 to get a probability of .9616.
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1answer
17 views

studying set I of population

let x be the mean and € the standard deviation of the statistics ×1,,,,,,, xn. let I=(x-3€, x+3×€) and the number of items not in I is k. prove that n greater or equal to 9k. prove that the ...
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19 views

Regression with many discrete and continuous predictors and few rows

I want to do regression on a dataset. It has one continuous dependent variable that I want to predict. It has many categorical and some continuous predictors. It only has a few rows. A simplified ...
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3answers
187 views

Is this a Conditional Probability?

I am very confused. Is the highlighted $ P(\text{Graduated} \cap \text{Studied})$ or $ P(\text{Graduated} \mid \text{Studied})$
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11 views

low number poissonian errors: is it possible for a measurement of 4 counts to have a significance higher than 2 sigma?

What is the best way to measure statistical significance of an overdensity in counting experiment where you have small numbers? Rather than bore you with my actual problem, please consider the ...
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1answer
44 views

Expected Value Proof - Law of Total Expectation.

Given that X and Y are random variables show that: $$E[E[X \mid Y]] = E[X]$$ I was thinking that I could use the definition of expected value (the summation one) to solve this, but when I tried I ...
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1answer
22 views

Shipment combinations/Hyper geometric distribution word problem.

My train of thought says: $P(X=1)=\frac {\binom {4}{1} \binom {19}{2} } {20 \choose 3} = 0.60$ This is solution cropped out from a paper. I believe the person made a mistake or most likely I did..
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28 views

Poisson Distribution Word Problem

The image you are looking is a solution to a problem that has been cropped out. I'm certain the solution is incorrect since it does not include P(X=2). Just to be on the overly safe side, I decided ...
2
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1answer
49 views

Urn problem and combinatorics

You have $5$ red and $4$ black balls. How many ways there are to distribute all to $3$ different bottles? If I had $9$ red balls, then it would be $\binom{n+k-1}{k}$ = $\binom{3+9-1}{9}$, but I have ...
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1answer
36 views

Tough Algebraic Simplification

I'm having an hard time solving the following expression. In order to get you tuned this is an extract of an Integral containing the multiplication of two Normal Distributions. This is very similar to ...
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2answers
44 views

Unbiased estimator.

With sample variance defined as $S^2 = (\sum_{i=1}^n (X_i - \overline{X})^2 )/(n-1)$ A. Show that $E(X_i^2) = \sigma^2 + \mu^2$ using the fact that $\sigma^2 = E((X_i - \mu)^2)$
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6answers
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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
20 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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1answer
19 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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21 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
26 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
63 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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0answers
41 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 ...
2
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2answers
86 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 ...
4
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1answer
59 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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53 views

Determining probability generating function for event “$SS$”

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
26 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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4answers
72 views

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

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
47 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
40 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
21 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 ...