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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23 views

Show that this is the correct decision rule on a likelihood ratio test

I am trying to solve the following question: Let $X_1 ... X_n$ be a random sample from the following PDF: $$\frac{\theta}{x^{\theta+1}}, x \ge 1, \theta \ge 0 $$ If $$p = \prod_{i=1}^n X_i$$ ...
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11 views

Determining Causal Relationships Between Two Processes (Pierce-Haugh)

Pierce and Haugh did some research on causality in temporal systems. For simplicity, consider two time series $\{X_{t}\}$ and $\{Y_{t}\}$. Suppose that both follow a causal and invertible $ARMA$ ...
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0answers
62 views

Cheating in a competition? [closed]

So this is based on something which is actually happening to my family right now. My dad operates a petrol station, and they have monthly competitions, I think, where for example you buy a slab of ...
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2answers
34 views

Why we consider log likelihood instead of Likelihood in Gaussian Distribution

I am reading Gaussian Distribution from a machine learning book. It states that - We shall determine values for the unknown parameters mu and sigma^2 in the gaussian by maximizing the ...
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14 views

We are trying to find the correlation between Stock Price index and Natural Disasters. I dont know what Statistical Method To use. [migrated]

Can someone help me identify what statistical method ( or any method ) that i can use to correlate the effects of natural disasters on Stock Market Index.
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2answers
73 views

$X = (X_1, X_2)$ is it not a multivariate random variable?

$X=(X_1,X_2,\ldots, X_P)$ is a $p$-dimensional random variable on $(\Omega, S, P) $ iff $X_i$'s are univariate random variables on the same probability space $(\Omega, S, P)$ ." We all know ...
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1answer
66 views

Proving $E_{\theta}[T(X)] = \frac{\psi'(\theta)}{\eta'(\theta)}$

I'm trying to understand how to prove the following theorem: Let $\{P_{\theta}, \theta \in \Theta\}$ be a family of distributions in the one parameter exponential family with density (pmf) ...
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14 views

Asymptotic Distribution of Cross-Correlation Function Between Two AR(1) Processes

Let $$X_{t} = \phi_{x}X_{t-1} + e_{t}$$ and $$Y_{t} = \phi_{y}Y_{t-1} + \epsilon_{t}$$ be two $AR(1)$ processes. My textbook writes that $\sqrt{n} \cdot r_{XY}(k) \sim N \left(0, \frac{1 + ...
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175 views

Why sample statistics converge to the right parameter

We know that for sample/empirical distribution function $F_n(x)$ we have that a) $F_n(x)\xrightarrow[p]{}F(x)$ (pointwise convergence) b) ...
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1answer
25 views

Pareto Distribution transformation

Suppose $X$ is a random variable with Pareto distribution. There pdf and cdf are: $$f_X(x) = \begin{cases} {\alpha x_m^\alpha \over x^{\alpha +1}}, & \text{if $x\ge x_m$ } \\ 0, & \text{if ...
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2answers
84 views

Binomial theorem combinatorics derivation

We can split any term $(x+y)^n$ into a choice tree with $2^n$ path choices. Also, we can represent each factor in terms of $x^ky^{n-k}$ It is logical to deduce we can express this with summation ...
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10 views

Determining Stationarity of (Multivariate) Processes

A necessary and sufficient condition for a vector autoregressive process to be stationary is for its eigenvalues to all have modulus less than one. I learned this result in the case of a model of this ...
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1answer
22 views

Matthews Correlation Coefficient yielding values outside of [-1,1]

(Note, this question has been cross-posted on the Cross-Validation site, which I wasn't aware of at the time of initial posting). I'm using the formula found on Wikipedia for calculating Matthew's ...
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0answers
32 views

What should the contestant do? [duplicate]

Suppose there are three curtains. Behind one curtain there is a nice prize while behind the other two there are worthless prizes. A contestant selects one curtain at random, and then one of the other ...
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1answer
21 views

How to make this bet fair?

A person bets $1$ dollar to $b$ dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. How to find the value of $b$ so that the ...
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1answer
57 views

Statistics, exponential distribution

I got the following task: The durability for a light bulb is following an exponential distribution with the expected value 800 hours. In a room there are 30 bulbs and all of them started to shine at ...
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1answer
21 views

How to compute of each player winning this sequence of games?

Players A and B play a sequence of independent games. Player A throws a die first and wins on a "six." If A fails, then player B throws and wins on a "five" or "six." If B fails, then A throws and ...
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3answers
40 views

How to compute these probabilities?

A pair of dice is cast until either the sum of seven or eight appears. How to compute the probability of a seven before an eight? Now, if this pair of dice is cast until a seven appears twice or ...
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1answer
35 views

Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$

Let $X_1, X_2, ...$ be a sequence of real-valued random variables. Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$ Attempt: Suppose ...
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2answers
47 views

How to compute this probability?

A drawer contains eight different pairs of socks. If six socks are drawn at random and without replacement, how to compute the probability that there is at least one matching pair among these six ...
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10 views

Showing Hat matrix equal specific values

Consider a one way layout model $y_{ij}$ = $\mu_i + e_{ij}$ (1 $\leq$ i $\leq$ a, 1 $\leq$ j $\leq$ $n_i$) where a = 3 and $n_1$ = 2, $n_2$ = 3, $n_3$ = 4. Show that the hat matrix for this design ...
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19 views

How to establish the independence or otherwise of these compound events?

Suppose that $C_1$, $C_2$, $\ldots$, $C_n$ are mutually independent events in a sample space $S$. Then how to establish the independence or otherwise of these combinations of events? $C_1^c$ and ...
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1answer
22 views

How to calculate individual chance over a total 10% of 60 iterations?

I'm developing a game that needs to calculate a 10% chance over a second of shooting against the player. The problem is, this calc is run every frame at a 60 frame rate. So I need the individual ...
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3answers
35 views

Expected value of n trials where probability of an event occuring is 1/n

Say the probability of an event occurring is 1/1000, and there are 1000 trials. What's the expected number of events that occur? I got to an answer in a quick script by doing the above 100,000 ...
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1answer
57 views

Why use regularization to reduce over-fitting

I'm having trouble understanding why should we use regularization for over-fitting when we can simply reduce the number of order to our polynomial function? Is it because it saves us time from having ...
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1answer
69 views

What is asymmetry index in an array of numbers?

Someone gave me this question , and I cannot find any source that can be helpful. Find the asymmetry index given an array of numbers and an integer -> find asymmetry index of the array Note: I ...
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1answer
18 views

How can I find the vector between two sets of data?

I need to identify the vector between two sets of data. The goal is to correctly "guess" whether a new piece of data is in group A or ...
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1answer
19 views

Find a standard normal probability

Given a standard normal distribution, find the area under the curve that lies to the left of $z=1.43$. It would make sense to me that I need to find $P(z<1.43$). Using a table that gives values ...
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0answers
29 views

How many data points from a sample are in the region $(X>F^{-1}_X (p),Y>F^{-1}_Y (q))$? [closed]

Assume that, there are two variables $X$ and $Y$, and there is is a sample consisting of 1000 observations $(x_i,y_i)$ , $i=1,2,\ldots,1000$, $p$ and $q$ are the quantiles of $X$ and $Y$ respectively. ...
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34 views

Determining $\sigma$ given mean and proportion of a Normal distribution?

The marks of a random sample of students with mean $\mu$ and standard deviation $\sigma$ showed that 15.87% scored higher than 70. The distribution of the marks is Normal with mean $50$ standard ...
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12 views

Statitical error and binning: data hard to interpret

I'm trying to understand by applying it the tool of binning samples of a dataset to extract the correlation time (between successive sample if they are generated by a Markov Chain for instance) from ...
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13 views

Fitting of the Lévy triple

Given a Lévy process and its triplet $(\mu,\Sigma,\nu)$ i.e. the triplet such that for each $t\ge 0$ $ X(t) = bt + W_A(t) + \int_{|x|<1} x \tilde N (t, dx) + \int_{|x|\ge 1} x N(t,dx)$ where ...
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25 views

probability that a variable, as a function of choice variables, is among the top k out of n when ordered

Suppose $(h_1,h_2,...,h_n)'$ is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a choice variable which can vary across $i$ and $X_i$ is a random shock with Pareto Type I distribution. ...
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1answer
35 views

Odds to guess a 32 byte value [closed]

I have 1,000,000 records, and each is assigned a 32 byte (3.4E+38) random value. What is the likelihood to guess one of the random values? Context This comes up in information security context: ...
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4answers
54 views

What is the probability that at least one letter is in the correct envelope?

A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct ...
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2answers
28 views

How many bulbs should be inspected for probability to exceed $1/2$?

In a lot of $50$ lightbulbs, there are $2$ bad bulbs. How many bulbs should be examined so that the probability of finding at least $1$ bad bulb is at least $1/2$? My effort: Suppose $n$, where $0 ...
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1answer
12 views

Choice of distribution given data

I have some data to be analyzed. It's histogram looks unimodal, with the support being positive reals between 0 and 100, most of the values huddled up around the mode.I want to be able to ...
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1answer
32 views

Probability Of a question [closed]

Three questions with a 50% chance to answer each correctly. What is the probability of answering at least one question correctly?
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1answer
36 views

Confidence interval for estimating probability of a biased coin

Suppose we have a coin with a probability $p$ of coming up heads and $q = 1-p$ of coming up tails on any given toss. (A coin is biased unless $p=0.5$). But we are not given what $p$ or $q$ are. We ...
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1answer
36 views

Board Game Score Calculation Suggestions?

My coworkers and I play board games at lunch. We've taken it so seriously that I've written an application to calculate a "player score" based on how you place in each game played. Here's how it ...
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1answer
17 views

Independence of two discrete random variables and their support.

I know that two discrete random variables, $X$ and $Y$, are independent if $P(X = x, Y = y) = P(X = x)\cdot P(Y = y)$. I also know that if the underlying space, $S$, of their joint pmf is ...
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0answers
15 views

What is the equation for an anisotropic Hanning window (cosine wave) in two or three dimensions?

I do not exactly know how to ask this question, so I will explain myself thoroughly. I am really stuck on this one, and it is crucial for my research, so if anyone has any ideas on where I may find ...
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1answer
37 views

Probability distribution for n-th smallest value in array with random values

I have a $d$-dimensional ($d>3$) array (vector, set, ...) which is filled with random values taken from uniform distribution (interval $[0,1]$). What is the probability distribution for n-th ...
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7 views

Sampling with an “oversampling” factor, in K-Means||

I'm trying to understand K-Means||, a scalable version of K-Means++, which itself is an "improved" version of the clustering algorithm K-Means. Please find here the link to K-Means|| paper ...
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35 views

Prove $T(e) = \int_{e}^{\infty} (y-e) \mathrm{d}F(y)$ is convex

Prove that $T(e) = \int_{e}^{\infty} (y-e) \mathrm{d}F(y)$ is a convex function, where $F(y)$ is cumulative distribution function of some real random variable $Y$. This is an exercise from ...
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1answer
26 views

How to rearrange multiple regression equation to solve for an independent in excel

Is there a way to solve this equation for $F_2$? I am using excel to do this if that matters. Thanks for any help. $Y=B_2+B_3F_2+B_4G_2+B_5H_2+B_6I_2+B_7J_2+B_8K_2$ ...
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1answer
31 views

Regression with Mean, Standard Deviation, Range and Correlation

A research team collected data on students in a statistics course. Their dependent variable was the student’s score on the final examination, which ranged from 200 to 800 points. The observed average ...
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2answers
38 views

Creating a polynomial function with no x-intercept

I have an understanding of polynomials and how to create a function based on the leading coefficient, degrees, x-intercepts, etc. My question is how do i go about creating a polynomial function that ...
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1answer
30 views

Sample Size for Correlation Testing

A research team wishes to test the null hypothesis: $H_0, r=0$ at $\alpha = 0.025$ against the alternative: $H_1, r>0$ using Fisher’s transformation of the Pearson product moment correlation ...
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1answer
36 views

Probability that a random variable is among the top k out of n when ordered

Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$. What is the probability distribution that any given $X_i$ is among the ...