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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1answer
245 views

ML estimate vs expected value?

I have a random vector $x=[x_1,x_2,...,x_n]^T$ with prior distribution Normal(0, I). I have $m$ linear constraints summarized in matrix form $Ax=y$ where $A$ is an $m$ by $n$ matrix and $y$ is an ...
4
votes
3answers
1k views

Integral with Normal Distributions

I know that the following equality is true for any $a$ and $\sigma$ (I have solved it numerically): $$\int_{-\infty}^{+\infty}\Phi\left(\frac{a-x}{\sigma}\right)\frac1{\sigma} ...
1
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1answer
601 views

poisson process an probability

Calls to an emergency ambulance service in a large city are generated by a Poisson process with λ = 3 calls per hour. How can I find the nature of the probability distribution for the lengths of time ...
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2answers
2k views

What is the standard error of the mean of an exponential distribution of the form $Ae^{Bx}$ with N measurements?

What is the standard error of the mean of an exponential distribution of the form $Ae^{Bx}$ with $N$ measurements?
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1answer
350 views

Plim of the expectation of the reciprocal of the sample mean of an autoregressive process

I am trying to prove the following: Let $X_t$ be a sequence of random variables that follows an autoregressive process; i.e. $X_t=X_{t-1}+e_t$, where $e_t$ is a zero mean i.i.d. sequence. Then ...
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2answers
237 views

expectation of incomplete gamma

Is the expectation of the (upper/lower) incomplete gamma function known? $$\int_0^{+\infty} x \Gamma(A, x) \mathrm dx$$
4
votes
1answer
202 views

Hypothesis testing

Suppose we have a random sample $X_1$, $X_2$ from the Beta($\theta$, $1$) distribution and we want to test $H_{\theta} :\theta \leq 1$ against $H_1:\theta > 1$. The following test issued: “Reject ...
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0answers
47 views

Finding a confidence interval for a parameter of regression given two variable variances

Given an equation $Y_i=0.1+0.3x_i+e_i$, I am asked to calculate a 95% confidence interval for $Y_i$ when $x=0$. So I have an equation $Y_i=0.1+e_i$, and I know that standard error of 0.1 is 0.005, ...
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2answers
851 views

Intuitive meaning of Pearson Product-moment correlation coefficient Formula

I can't understand the intuition behind Pearson Product-moment correlation coefficient Formula for bivariate data. The formula is : $\rho$ = cov(X,Y)/($S_x$ * $S_y$) where cov is covariance. $S_x$ and ...
3
votes
2answers
113 views

Computing the population given a set of replication rates

So I'm looking to calculate the probability that a bacterial population dies out with the following conditions: Initial population is 1 At each iteration the bacteria can die, do nothing, divide ...
4
votes
2answers
254 views

Confidence level in answer being correct when multiple people give that same answer

We are very rusty on our math, probability, statistics, permutations and combinations and need some help figuring out how to calculate some relative "confidence level" based on the number of people ...
1
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3answers
505 views

$\mathbb{E}(|X-Y|^3)$ Absolute expected value

I need to find $\mathbb{E}(|X-Y|^3)$ where $X$ and $Y$ are independent distributions and are continuously uniform distributed on interval $[0,1]$.
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2answers
449 views

Is this true: $\mathrm{Cov}(X,Y) = \sqrt{\mathrm{Var(X)}}\sqrt{\mathrm{Var}(Y)}$?

I just want to confirm whether this is correct or not: The covariance of $X$ and $Y$ is equal to the standard deviation of $X$ times the standard deviation of $Y$. or, in mathematical notation, ...
-1
votes
1answer
613 views

Conditional Expectation, discrete

Let X1 and X2 be the numbers on two independent fair-die rolls. Let X be the minimum and Y the maximum of X1 and X2. Calculate: $$E(Y|X=x)\qquad\text{and}\qquad E(X|Y=y) $$ given that X1 and ...
2
votes
2answers
119 views

Is there a mathematical name given to the count of negative and positive numbers in a set

If I have a set of numbers {-1, 2, 3, 4, -8, 2, 0, 44} and I make the statement that there are: 2 negative numbers 5 positive numbers and one signless number Is there a mathematical concept used to ...
4
votes
1answer
141 views

Given a set of numbers, what mathematical name is given to the most frequent number

Assume a set of numbers {0,1,2,3,4,4,4,4,4,5,6,7,8,9} What mathematical name is given to the number that most frequently occurs? For example, if I was to count the number of occurrences for each ...
0
votes
1answer
539 views

How do I fit a model with piecewise linear regression

I have a set of points in 3D (x,y,z). I ordered these points from the lowest to highest. So, I want to used linear regression to fit a line through these ordered points and then to find out a break ...
0
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1answer
260 views

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 ...
0
votes
1answer
778 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 ...
3
votes
2answers
188 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 ...
1
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1answer
177 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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votes
2answers
523 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 ...
0
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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
566 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 ...
6
votes
1answer
580 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
117 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 + ...
2
votes
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 ...
4
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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 ...
1
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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 ...
3
votes
1answer
206 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 = ...
2
votes
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. ...
3
votes
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: ...
5
votes
1answer
3k 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
405 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
44k 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 ...
1
vote
1answer
340 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 ...
2
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2answers
438 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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vote
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 ...
0
votes
1answer
622 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
573 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 ...
4
votes
1answer
669 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 ...
0
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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
295 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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3answers
343 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 ...
2
votes
2answers
454 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 ...