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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Probability - predicting the number of children in a small town

The first part of this question states: Consider a very small town with 50 families with children. Let X be the number of children in a family picked at random from the 50 families with children in ...
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1answer
71 views

Can one prove $\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min(1,\frac{c}{t})$?

Let $c>1/2$ be an arbitrary big fixed constant. Can one prove that for all $t\geq 1$: $$\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min\left(1,\frac{c}{t}\right)$$ for some small constant ...
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1answer
794 views

Maximum Likelihood Estimator for Multivariate Bernoulli

I am working on deriving Naive Bayes for document classification. Each document is represented by a binary vector $x^i$ where $i=1,..,N$ for N documents. In this vector a cell is set to 1 if that ...
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2answers
125 views

Explanations on matrix transposition

I've never really worked with matrices so I would be glad about some information on the following issue: In some statistic calculations there is often a transposed matrix within a formula. Can ...
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2answers
189 views

A variation on the $F$-distribution

If I have $\frac{X/n_1}{Y/n_2}$ where $X$ and $Y$ are independent chi-squared random variables, with degrees of freedom $n_1$ and $n_2$, respectively, then the distribution of this ratio is given by ...
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1answer
182 views

bound of Erlang distribution

Is there any known polynomial bound of the Erlang distribution? I'd like to say that, given $k$ and $\lambda$ with probability p the r.v. is going to be less than some value x.
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2answers
529 views

Buckets of Balls, Will one fill if I add another Ball?

I was refereed here by stackoverflow.com. With some searching I found this: another balls and bins question, but its not quite what I am looking for. Rather the inverse. IE the expected number of ...
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1answer
53 views

How can I calculate max value, if I know just number of records, minimal value and average value of all records?

How can I calculate max value or all possible and relevant maximum values, if I know number of records, minimal value and average value of all records? For example: Number of records (persons): 92 ...
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2answers
571 views

Conditional Variance

X is a random variable with mean 100 and standard deviation 30. Y is a random variable with mean of 50 and standard deviation of 25. The correlation between X and Y is 0.5. What is Var(Y|X)?
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3answers
1k views

Statistics, Poisson distribution + hypotheses

Hey, I find this problem quite difficult to handle,I would be grateful to anyone who could at least lead on the path of solving it. The random variable X has a Poisson distribution with mean µ. The ...
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1answer
587 views

Symmetric matrix decomposition with orthonormal basis of non-eigenvectors

I like to understand the following transformation found in documentation for deriving Kalman filter. Abstract Formulation: Given 2 symmetric matrices $A$ ,$B$ $\in$ $\mathbb R^{3,3}$ with $A \ne B$ ...
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1answer
118 views

Transformations of Normal

If we have $U$, $V$, and $W$ as i.i.d normal RV with mean $0$ and variance $\sigma^2$, then what are the following expressed as a transformation of a known distribution (if known): 1) $\frac{U}{V + ...
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2answers
2k views

Bivariate Normal Conditional Variance

I am given the parameters for a bivariate normal distribution ($\mu_x, \mu_y, \sigma_x, \sigma_y,$ and $\rho$). How would I go about finding the Var($Y|X=x$)? I was able to find E[$Y|X=x$] by writing ...
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1answer
4k views

What exactly is the Probability Integral Transform?

I've been going back over my notes from Stats class and came across the Probability Integral Transform. From my limited understanding, the basic idea is that from a cdf in terms of one variable, can ...
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1answer
117 views

Geometric Distribution #2

a TV show tests a number of people who claim to be "psychic". The test involves blindly predicting each outcomes of 5 rolls of a fair die. The TV show will declare such a person as psychic if they ...
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1answer
210 views

Variational Distance vs. maximum norm

Suppose I have vector $x^t \in \mathbb{R}^n, x_i > 0$ that is a random variable in $t$. I define a measure $D(x) := \max_{i,j} |x_i - x_j|$, which essentially is the maximum discrepancy of any two ...
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0answers
1k views

Calculate relative contribution to percent change

Let me use a simple example to illustrate my problem. First, assume we are calculating rate r at time t such that rt = xt / yt. Furthermore, each measure has two component parts: X = xa + xb and Y = ...
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2answers
237 views

Density Question - Statistics

A point is picked randomly in space. Its three coordinates $X$, $Y$, and $Z$ are independent standard normal variables. Let $R = \sqrt{X^2+Y^2+Z^2}$ be the distance from the point from the origin. ...
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2answers
2k views

CDF of a ratio of exponential variables

Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$. I tried out the problem, and wanted to check to see if my answer of: ...
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1answer
4k views

Probability of duplicate GUID

A GUID (globally unique identifier) is a 32 character hexadecimal string: http://en.wikipedia.org/wiki/Globally_Unique_Identifier If you randomly generate 2, the chance of them being the same is ...
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1answer
406 views

Geometric Distribution

Let X be a random variable with the geometric distribution. For m<n, what is P( X > n| X > m)?
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6answers
45k views

How to calculate percentile? Is it possible to get 100 percentile?

How do we calculate percentile? I think it should be calculated as: P = Total number of candidates L = Number of candidates whose marks are below yours ...
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1answer
353 views

Distribution Question Stat

In a particular town 10% of the families have no children, 20% have one child, 40% have two children, 20% have three children, and 10% have four. Let T represent the total number of children, and G ...
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2answers
444 views

For a covariance matrix, what would be the properties associated with the eigenvectors space of this matrix?

I want to know, since the covariance matrix is symmetric, positive, and semi-definite, then if I calculate its eigenvectors what would be the properties of the space constructed by those eigenvectors ...
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1answer
3k views

Why do we subtract the variance?

This is not a question for doing my homework. This is a question to understand the deeper meaning of the answer. So in part b), it subtracts the variance. Why do we subtract variance and what does it ...
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0answers
107 views

How to interpret the sum notation on $\sum_{r:i(r)=i} \langle R_{r\alpha} \rangle^{(t)}$

While doing research for my thesis, I ran into a paper called "Statistical Models for Co-occurrence Data". In the early pages, when talking about an iterative numerical method (a custom EM-method, to ...
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1answer
625 views

Independent Normal Variables

Let X and Y be indep. standard normal variables. Find: a) P(3X + 2Y > 5): This is just 1 - phi(5/sqrt(13)) from the fact that the mean is 0, and the std. deviation is sqrt(13). DId you get 0.0838 as ...
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3answers
581 views

Linear conditional expectations example

I am looking for an example of two random variables $X,Y$ such that (a) $X,Y$ are not independent. (b) At least one of $X,Y$ is not normal. (c) $E(X|y)$ (expected value of $X$ given $Y=y$) is ...
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1answer
684 views

How can I get better at this certain kind of probability problem?

I'm studying for an onsite interview with Google for a product manager position. While looking at interview questions online, I've realized that I really need to brush up on the probability and ...
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1answer
366 views

Probability with Independent Normal Variables

Let X and Y be independent standard normal variables. Find: a) P(3X + 2Y > 5)
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2answers
296 views

Uniform $(-1,1)$ distribution

Let $X$ and $Y$ be independent with uniform $(-1,1)$ distribution. Please help in finding: a) $P(X^2+Y^2 \leq r^2)$ b) The CDF of $R^2 = X^2 + Y^2$ c) The density of $R^2$ All I tried was breaking ...
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345 views

Uniform Distributions - Probability

Suppose $U_1$, $U_2$ and $U_3$ are independent uniform $(0,1)$. I am supposed to find $P(\max(U_1,U_2) > U_3)$. What I rewrote the question as was this is equal to: $$2P(U_1>U_3) - P(U_1 ...
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2answers
475 views

probability distribution of coverage of a set after `X` independently, randomly selected members of the set

I have a set of numbers where I am randomly and independently selecting elements within a set . After a number of these random element selections I want to know the coverage of the elements in the ...
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1answer
138 views

Interpolation needed

I have a series of datasets which I need to interpolate, I did this once in uni but that was a long time ago. I could use any pointers I can get. So I have put the data up here in the hope that ...
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1answer
235 views

If the covariance of A, B if E(A) = 0, does convariance of A, B = 0?

So I have got 2 variables, $A$ and $B$. I know for a fact that $E(A)$ = 0. I don't know if they are independent. $$Cov(A, B) = E(A B) - E(B) E(A)$$ I know that $E(A) = 0$. Does that mean $E(A B) ...
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2answers
236 views

What is the difference between $\mathbf X\mathbf X^\prime$ and $(\mathbf X-\mu)(\mathbf X-\mu)^\prime$?

In $\mathbf X\mathbf X^\prime$, $\mathbf X$ is a matrix contains data points in column fashion, $\mathbf X^\prime$ is its transpose, this looks like a covariance matrix, but does not subtract mean, so ...
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2answers
260 views

What's the proper function to calculate confidence percentages for Chi Squares?

In the book Practical BASIC Programs by Lon Poole (Osborne/McGraw-Hill, 1980) (~10M pdf), the sample program listing has a number of discrepancies and some outright incorrect statements. On page 143 ...
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0answers
537 views

Use of covariance matrix for the confidence interval

I have a number of explanatory variables $x_1,...,x_n$ and an outcome variable $y = f(x_1,...,x_n)$. Here $f$ is assumed to be known (estimated). I heard that for a confidence interval for $y$ one can ...
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1answer
597 views

What's the probability that a bookmaker is mispricing odds on soccer games

An English soccer team plays a series of matches again different opponents, of varying ability. A bookmaker offers odds for each match as to whether it will be a home win, away win, or draw. Part-way ...
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1answer
251 views

Can I build a program that will tell me if a real world data set looks linear, logarithmic, exponential etc?

I have a bunch of real world data sets and from manually plotting some of the data in graphs, I've discovered some data sets look pretty much logarithmic and some look linear, or exponential (and some ...
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2answers
289 views

The Metropolis Algorithm

I Know how to apply the Metropolis Algorithm, but I'd be grateful if someone could explain to me the reasoning behind the steps in the algorithm. I've tried in vain looking for the original paper. ...
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1answer
107 views

Determining the distribution of a population from a sample

I have a uniformly collected sample of 10000 data points from a population of about 200000. I'd like to find out what the distribution of the population is. How can I do this rigourously?
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3answers
382 views

Usefulness of Variance

I've had a look for intuitive explanations of the variance of an RV (e.g. Intuitive explanation of variance and moment in Probability.) but unfortunately for me, I still don't feel comfortable with ...
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1answer
512 views

Time Until Extinction for a Pure Death Process Where the Time is Exponentially Distributed

Let $X(t)$ be a pure death process starting from $X(0)=N$. Assume that the death parameters are $\mu_1, \mu_2,\dots,\mu_N$. Let $T$ be an independent exponentially distributed random variable with ...
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2answers
550 views

How to compare randomness of two sets of data?

Given two sets of random numbers, is it possible to say that one set of random numbers has a greater degree of randomness when compared to the other? Or one set of numbers is more random when compared ...
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101 views

Probability and $Z$-Scores

I'm new here and was hoping you guys could help me with a statistics problem that I don't quite understand. I'm not sure if it's proper etiquette to ask for help on a specific homework problem here, ...
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2answers
115 views

What are the conditions under which a distribution reduces down a dimension?

What are the conditions under which a distribution reduces down a dimension? For example, suppose I have a 2D gaussian distribution for X and Y. Under what condition(s) on Y does the distribution ...
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1answer
62 views

Variance of a recursed substition function

So, I have this function: $$ y_t=v_t+\rho v_{t-1}+\rho^2 v_{t-2}+\dots+\rho^{t-1}v_1+\rho^ty_0 $$ And I want to find the variance (and after that the covariance, but I should be able to do that..). ...
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3answers
802 views

Can you demystify the Power Law?

How would you describe the Power Law in simple words? The Wikipedia entry is too long and verbose. I would like to understand the concept of the power law and how and why it shows up everywhere. For ...
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1answer
1k views

leading and lagging moving average indicator

What are leading short and lagging long moving average indicators and how do we calculate them? e.g. for the following data set, and let window size be 2. Can you ...