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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11 views

Find the required Chi-square score for an arbitrarily low p-value (2 degrees of freedom)

I'm trying to use the Chi-Square test to find the significance of data that suffers from the multiple testing problem. Because I have this multiple testing problem, the required p-value to view a test ...
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7 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X)$ where $K = ...
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1answer
14 views

Slow convergence simulating log-normal sample from the normal

I am trying to simulate a log-normal random variable $Y$ with mean $m = \mathbb{E}[Y] = 0.001$ and standard deviation $s = 0.094$ by simulating a normal sample instead, and then exponentiating it. ...
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0answers
11 views

Departure from uniformity in a continuous (time) distribution

I know how to quantify the departure from uniformity ( or a uniform distribution) for discrete distributions. Assume you have a distribution set of P: ...
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0answers
8 views

Is Autocorrelation computed with cross-correlation or correlation?

I have seen multiple articles online that say that Autocorrelation is computed using correlation and cross-correlation. Which is autocorrelation actually computed with?
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1answer
20 views

Probability of a user references in a network [on hold]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
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2answers
14 views

statistical significance in probabilities

Imagine I am conducting an experiment, and I record whether $n$ individuals of different nationalities, say $A$, $B$, and $C$, either like or dislike a product. In the end I have the respective ...
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0answers
13 views

Proof of a classical Theorem of Martin-Löf on complexity dips for Kolmogorov complexity,

I have a question on the first Theorem from the article Complexity of Oscillations in Infinite Binary Sequences by P. Martin-Löf, which could be downloaded from the publisher or from here. Theorem ...
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1answer
42 views

Binomial Distribution Confidence Interval

Normally when I see confidence intervals it is in attempt to estimate a population parameter (probably poor wording). What I am trying to do is form a confidence interval for some theoretical values, ...
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1answer
17 views

computing weight from distance metric

I have a distance between two points in meters. I want to convert this distance into weight such that as distance increases the weight decreases. What are some good weighting function that can ...
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1answer
14 views

How to calculate a posterior probability with a given Gaussian Mixture Model?

I'm building a GMM-based classifier in speech processing and I'm using GMM as a probabilistic scoring mechanism (therefore I don't intrinsically care about the underlying mixture components). For ...
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1answer
8 views

How do I solve this question using Z Table and Normal distribution?

A company pays its employees an average wage of 15.90 an hour with a standard deviation of 1.50. Assume the wages are approximately normally distributed. a) what proportion of employees receive ...
1
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1answer
17 views

How do I use interpolation with the Z table?

My textbook has an example of interpolation, but I am not sure how the book did it since it doesn't explain it. It says if we want $P(Z < 1.246)$ we must use interpolation and the steps given are: ...
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2answers
79 views

Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...
2
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1answer
23 views

Intuition on Martin-Löf-Test for finite strings

The followng example is from An Introduction to Kolmogorov Complexity and Its Applications, Example 2.4.1. and is concerned with Martin-Löf-Tests for finite strings: A string $x_1 x_2 \ldots x_n$ ...
3
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1answer
24 views

Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi _{2n}^2$.

The Statement of the Problem: Suppose that $X_1,\ldots, X_n$ is a random sample from the $U(0,1)$ distribution and $$ Y_i = -2\log X_i. $$ Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi ...
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1answer
25 views

How to find median from a probability distribution?

Having trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...
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2answers
18 views

Difference between the two ways of finding average winning rate over a period of time

Ok I might be asking something stupid, please bear with me, math is not my strong suit. This has to be a duplicate but I dont know how to search them. I dont even know how to word the title, kindly ...
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2answers
22 views

Show that the conditional pmf of $X_i$, given $T = t$, is $Binomial(t, \lambda_i/\lambda).$

The Statement of the Problem: The random variables $X_1, ..., X_n$ are independent and $X_i \sim Poisson(\lambda _i), i = 1, ..., n$. Set $$ T = \sum_{i=1}^n X_i \qquad \text{and} \qquad \lambda = ...
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0answers
25 views

Pure death processes

If $P_n (t)=\Pr (N (t)=n)$ and $N (0)=a$, how can I show that in a pure death process $$P_{(a-1)}(t)=a (e^{\mu t }-1)e^{-a \mu t}.$$ I showed that $P_a(t)=e^{-a \mu t}$. In fact I want to show ...
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2answers
30 views

Central Limit Theorem and understanding mean for a single object

The IQ of actuarial science majors is assumed to be normally distributed with mean 112 and standard deviation of 14. In a class of 19 students, find the probability that the mean IQ of all 19 students ...
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2answers
29 views

Variance of |X-Y| for X and Y ~ N(0,1/2)

I know $X$ and $Y\sim\mathcal{N}(0,\frac12)$, $X$ and $Y$ are independent. I try the following way to solve variance of $g(X,Y)=|X-Y|$ ,which is $V(|X-Y|)$. If ...
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1answer
26 views

Probability of time between two events in a poisson process

Suppose people arrive at a certain place according to a poisson process with rate 10 per day. 1) What is the expected time until the arrival of 100 person. 2) What is the probability that ...
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0answers
15 views

Hamiliton Circuit [on hold]

Consider the graph given above. Use the sorted edges algorithm to find a Hamiltonian circuit. a. List the weights of the edges separated by commas in the Hamiltonian circuit in the order they are ...
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4answers
147 views

How to understand $E(XY)$ intuitively

I have no trouble understanding $\displaystyle E(X)=\int xf(x)\,dx $ and $\displaystyle E(Y)=\int y f(y)\,dy$ As each $x$ multiplies the corresponding $f(x)$ and we take the integral of it to ...
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1answer
15 views

Deriving simple linear regression from normal equations

The normal equations of least squares regression $$X \beta = Y$$ yields the solution $\beta = (X X^T)^{-1} X^T Y$. For simple (1-dimensional) regression, the solution to $\beta x_i + \alpha = y_i$ is ...
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1answer
12 views

Recover true statistics for a union of subsamples - only data available are summary statistics for each subsample

Let $a_1$, $a_2$, ...,$a_m$ be the samples of data, and let us further assume the only information we have about each sample is their count/no. of samples, mean, standard deviation and median. The ...
2
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1answer
17 views

Measuring correlation of a truncated sample

Suppose samples $(X_i,Y_i)$ were drawn from a multinomial distribution $N(\mu_X, \mu_Y, \sigma_X, \sigma_Y)$. The correlation between $X$ and $Y$ can be then estimated as ...
2
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1answer
36 views

Conditional distribution of $X$ exponential given $U\leq e^{-X}$, with $U$ uniform on $(0,1)$

Let $X$ be exponentially distributed with mean $1$ and $U$ be a $U(0,1)$ random variable independent of $X$. Define $$I= \begin{cases}1,&U \leq e^{-X}\\ 0,&\text{ ...
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3answers
39 views

Trying to understand Bienaymé formula

In Bienaymé formula, it states that $var(\bar X) = \large\frac{\sigma^2}{n}$. However, when I was going through the proof here, it says the variances of $X_1,X_2,X_3......X_n$ are the same(assuming ...
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3answers
69 views

The probability of two consecutive non-leap years having 52 Fridays each is $\frac{5}{7}$. How?

I took a test on probabilities, and there was this question about finding the probability of two consecutive non-leap years having 52 Fridays each. I figured it would be $\frac{6}{7} \times ...
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0answers
18 views

Confidence interval of median of log-normal distribution

Assume 101 families ($n$) were questioned about their spendings in the month of July, and the amount of money they earn. Say $X$ is the difference in monthly income and their expenses in July. This ...
2
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1answer
37 views

Independence, conditioning, and correlation part 2 [on hold]

Suppose $X$ and $Y$ are independent random variables. I now want to consider conditioning on some event $C$. Under what conditions will $X\mid C$ and $Y\mid C$ be correlated?
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2answers
29 views

Variance of least square estimator

I have two random variables X and Y with $X\sim Exp(a)$ and $Y \sim Exp(\frac a2)$. I have a least square estimator $a=\frac {2x +y}{2.5}$. I want to calculate the variance of the estimator and to do ...
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1answer
35 views

Probability of decision on arrival of “new” information

I just decided to share a new problem (Rice's book, problem 1.58) and supposed solution: A teacher tells three boys, Drew, Chris and Jason, that two of them will have to stay after school to help ...
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2answers
18 views

Is a nondescript shopping statistic a normal distribution?

Suppose we have a certain product for which you can count how many times it's been bought every week for an entire year. Suppose I made a graph where every index on the X axis illustrates how many ...
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0answers
20 views

statistics- people's shopping [on hold]

I am building some project (website) on people's shopping (stuff stored in refrigerator) . I need to generate an automatic shopping cart based on people's shopping . For example if some person buys ...
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0answers
28 views

maximum likelihood estimation econometrics

Consider a model $$y=\beta_0+\beta_1x+\epsilon$$ And the $\epsilon$ has a following distribution. $$p(\epsilon_i)=\frac{1}{2\theta\cdot \exp(|\epsilon_i/\theta|)}$$ Find the maximum likelihood ...
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2answers
46 views

conditional probability of coming up 3 at least in one dice

Sorry for the simple problem, but I wanted to make sure that my idea is right (this is is the problem 1.8.50 from Rice's book) Two dices are rolled and the sum of the face values is six. What is the ...
2
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4answers
86 views

When will Andrea arrive before Bert?

The question was as follows- on any given day, Andrea is equally likely to clock in at work any time from 8:50am to 9:06am. Similarly, Bert is equally likely to to clock in at work at any time ...
0
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2answers
35 views

Normally distributed rv

Here is a question from a past statistics test in my university: An automatic filling machine is to pour beer in $340 \ c.c.$ bottles . The amount poured $X$, is a normal variable having a mean of ...
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0answers
17 views

How to check if a factor does affect a variable?

I have an indicator from a data set, described by a variable (e.g. Y). based on the general understanding, Y is affected by two possible factors, for example, factor a and factor b. however, i do not ...
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1answer
23 views

Basic Statistics problem 2 [on hold]

What is the difference between descriptive and inferential statistics? How to identify from a sentence the difference of the two?
2
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1answer
33 views

Poisson(Exponential) Distribution question

I am trying to solve the following question: If the number of calls received per hour by an answering service is a Poisson random variable with rate of 6 calls per hour, what is the probability of ...
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0answers
10 views

Extreme value distributions of unaccountably infinite set of random variables

Let us suppose that we have an uncountably infinite set $A=\{x_1,x_2, \cdots\}$ of i.i.d. random variables $x_i$, say with gamma distribution. Are minimum and maximum extreme value distributions ...
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0answers
19 views

Statistical significance of difference between samples given total success rate when test was applied by group

I am comparing treatments that select members to receive some email, and we are measuring the response to that email. The emails are similar, but are sent to a different group of members and a ...
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0answers
12 views

how to find the corelation or regression between two tables [closed]

I have two tables , I need to find the dependence or relation between these two tables table one ... $3$ sport v/s four regions with value as sales table two .. same $3$ sport v/s same four regions ...
0
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1answer
26 views

Partition-based entropy of a sequence

The entropy $H$ of a discrete random variable $X$ is defined by $$H(X)=E[I(X)]=\sum_xP(x)I(x)=\sum_xP(x)\log P(x)^{-1}$$ where $x$ are the possible values of $X$, $P(x)$ is the probability of $x$, ...
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0answers
13 views

Most appropriate ranking system

Imagine a system has N set of attributes, all with values of 1-20 (20 being desirable). There are a set of tests which can be performed on the system, each assessing a unique subset of the attributes. ...
0
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1answer
19 views

Confidence Interval w/ true standard deviation?

I'm very scared that my calculations I did were wrong. Here is why: I assumed true standard deviation meant population S.D. However the question says the standard deviation is from a sample. So what ...