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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What is the bound on $E\|Y_n\|^4$ in terms of $n$?

Let $X_n,n\in\mathbb{N}$ be i.i.d. zero-mean random variables in some separable Hilbert space with $E\|X_n\|^8<\infty$ and $Y_n=\frac{1}{n}\sum_{i=1}^nX_n$. I need to find bounds on $E\|Y_n\|^4$. ...
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3answers
4k views

1/1000 chance of a reaction. If you do the action 1000 times, whats the new chance the reaction occurs?

A hypothetical example: You have a 1/1000 chance of being hit by a bus when crossing the street. However, if you perform the action of crossing the street 1000 times, then your chance of being ...
2
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1answer
80 views

Monte Carlo p-test and early stopping

Say you have a coin with some probability $p$ of falling on heads. You would like to determine if this probability is less than or equal to $0.05$ with some reasonable degree of confidence and stop ...
2
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1answer
38 views

Expected Value on code

I'm trying to figure out the expected number of times this algorithm will print. I'm stuck on how to go about doing so. I used an indicator variable to keep track of the number of print statements ...
2
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1answer
53 views

Using the inverse Gaussian integral to find percentiles

I need some help with the following: Let $$R=\mu+\sigma*\epsilon \hspace{1cm} \epsilon \sim N(0,1)$$ I want to argue that $$ \mu + \sigma*\Phi^{-1}(u)$$ are the percentiles of the model when ...
2
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1answer
115 views

Hypothesis Testing help

Really have no idea where to start :( In an experiment comparing two weight-loss regimes A and B 20 test subjects were matched into 10 pairs so that within each pair the subjects were as similar as ...
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1answer
55 views

Cumulative probability of Chi-squared distribution

If $X$ is distributed $\frac{\chi_{10}^2}{10}$ , find the probability that $X > 1.83$ The formula for the Chi-squared CDF I'm using is the following, which is the integral of the PDF formula: ...
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0answers
44 views

Finding a sufficient statistic for an iid sample of the Gumbel distribution

$G(x;\alpha, \beta) = \exp\{-\beta e^{-\alpha x}\}$ for $x \in \mathbb{R}$ is a distribution (Gumbel family). Side question: is $G(x;\alpha, \beta)$ a member of the exponential family? I do not think ...
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1answer
61 views

Finding probability given mean and standard deviation

I don't know how to approach this problem: X is normally distributed with a mean of 200 and a standard deviation of 10. Find P(X ≥ 203)
3
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1answer
59 views

Generating random variables with complicated probability distribution functions

I have an interesting question I need to solve, and as much as I try, I cannot wrap my head around it. Given a postive random variable X with p.d.f. ...
0
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1answer
37 views

confidence Interval of multivariate Gaussian Distribution

Suppose I have an single variable which is gaussian distributed (with mean value 0 and standard deviation sigma). Then I certainly know that there is 68.2% of the chance that this variable should lie ...
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1answer
44 views

How to estimate the mutual information

I have two discrete non-negative random variables $X$ and $Y$. I know $X$ is the number of heads you get by tossing $n$ unbiased coins and I know $Y$ is in the range $0,\dots n$. I can sample from ...
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0answers
26 views

Stat problem! Why is this? [duplicate]

This is a statistics problem. although this is not a problem which needs an answer, I want to know the reason Why this is right. Can you guys help me ? Thanks in advance!
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0answers
27 views

Likelihood Functions of Nonparametric Simple Regression

I'm trying to find the likelihood function of a nonparametric simple regression model. Nonparametric statistics is new to me however, so I'm having some trouble wrapping my head around some of the ...
2
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2answers
29 views

Probability of 2 sets of triples in a 6 card hand

The deck is a standard 52 card deck. My solution was: The first card drawn can be anything, so 52 possible cards. The next card has to be the same value, so there are 3 possible cards. The last card ...
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1answer
87 views

Find the cdf associated with each pdf (NOT transformation)

Find the cdc associated with each pdf: a) f(x) = 3(1-x)^2 , 0 < x < 1 , zero elsewhere b) f(x) = 1/x^2 , -infinity < x < infinity The answers are a) 1-(1-x)^3 , 0 <= x < 1 b) 1 ...
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0answers
37 views

Probability concept applied to DNA sequencing.

Q: If an infinitely long DNA sequence is observed, can you determine the probability that an A occurs before G? You can recast this problem to ignore or eliminate all C's and T's. My attempt: I ...
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1answer
53 views

Finding an expression for a joint probability if two random variables have the same distribution function.

If $X$ and $Y$ are independent random variables with the same distribution function, say $F$, find an expression for $P(X<t, Y<t)$. My attempt: $P(X<t, Y<t) = P(X<t)P(Y<t) = ...
2
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1answer
36 views

How to compute N th Values for the series

I have 3 variables X, Y, Z $X_0=3$. $Y_0=1$. $Z_0=0$. $X_n=X_{n-1} + 3 * Z_{n_-1}$. $Y_n=X_{n-1} + 2 * Z_{n_-1}$. $Z_n= 5* Y_{n-1} $. I have tried alot to get a series ...
0
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0answers
29 views

Convergence of quantiles

Suppose that Fn is a sequence of distribution functions converging weakly to a distribution function F. Let Q(p) = inf {x : F(x) >= p} denote the pth quantile of F, for 0 < p < 1. Is it true, ...
2
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1answer
49 views

Help with financial modelling

We deal with "jobs" - a mini project involving multiple parties, resources and time constraints, etc. Jobs progresses through 4 well defined states from start to end; call them S1, S2, S3 and S4 where ...
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1answer
50 views

More variables = better fit?

When fitting (let's say) a linear regression model, it is always true, that the more variables we include in our model, the better fit is (in R^2 sense)? I don't want to discuss here overfitting, ...
3
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2answers
317 views

Expected number of output letters to get desired word

I am using a letter set of four letters, say {A,B,C,D}, which is used to output a random string of letters. I want to calculate the expected output length until the word ABCD is obtained; that is, the ...
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2answers
375 views

Chance on winning by throwing a head on first toss.

Problem: Players A,B and C toss a fair coin in order, the first to throw a head wins what are their respective chances of winning? Attempt: Let X = event that A throws a head on the first toss, and Y ...
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1answer
52 views

List of well-known submodular function in physics, statistics, math?

Can you please share a list of well-known submodular functions (have the diminishing return property) that you know? In physics, stats, math, etc? I am searching for a submodular function for my ...
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1answer
36 views

Definition of Random Sample in Estimation

In my statistics class, we're just beginning to talk about (point) estimation. I understand the basics for the most part, but I have a small question that might actually be due more to notation than ...
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4answers
36 views

Proving consequence of $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$

How to prove that if $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$, then $X$ and $Y$ are independent?
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1answer
17 views

Related to chi-squared functions

I'm finding difficulty in finding what type of function it is in continuous distributions in probability. Mainly how can I identify whether a function is chi-squared or not?
2
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1answer
153 views

Real life math to explore/solve

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
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0answers
24 views

Question about Lagrangian Multiplier (Gradient) Statistic of constrained GMM

I am trying to derive the Lagrangian multiplier statistic (GMM version) under a restriction. The question is given below The quadratic form is given by $Q_n(\theta,\alpha)=[m(\theta)', ...
2
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0answers
46 views

Differential Equation for brownian bridge?

For the brownian motion, we know that probability density of the particle's position at time $ t $, $ \rho(x,t) $ satisfies the diffusion equation pde: $ \partial_t \rho = d \; \partial_x^2 \rho $. Is ...
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0answers
33 views

How can I compute this?(method) [duplicate]

$$\mu = \sum\limits_{k=0}^{\infty}\dfrac{2^{k}}{\displaystyle{2k+1 \choose k}}$$ The answer is $\dfrac{\pi}{2}$. But I don't know how to do it. Please show me the way! Thank you.
2
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1answer
58 views

How to draw Congressional districts to mirror the Popular Vote

Let me preface this by saying that I'm not sure whether this is fundamentally a mathematical question or not, but I think it is. In the United States, the House of Representatives is elected roughly ...
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2answers
30 views

Permutations and Sample Spaces

Suppose 3 cars can either turn left $(L)$, turn right $(R)$, or go straight $(S)$. I need to find the sample space for all the possibilities but I am not sure how to do that. I know that for 3 cars ...
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0answers
102 views

balls in bins — waiting time until $k$ bins are occupied

Consider the classic balls in bins problem: we throw balls one by one into $n$ bins independently and uniformly. Define $\tau(k)$ for $1 \le k \le n$ to be the number of balls we have thrown until $k$ ...
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1answer
38 views

Under what assumptions can one compute conditional probability as $p(x)/p(y)$?

Conditional probability is often introduced in the following way: Consider a normal, fair 6-sided die. If you toss it then the probability $p(x=2)=1/6$. Now given that we already observed that the ...
0
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1answer
21 views

Pseudo-algorithm for most equals group size

Let's say I don't know how many (let's says person) will be present. I know I want to divise all those persons in group of 15. What king of algorithm could I use to create groups of person (the most ...
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0answers
47 views

MAP for exponential function (Maximum a posteriori)

I am trying to find the MAP for an exponential function of the form $p(y) = \theta.e^{{-\theta}y}$ Given that $\theta$ is constant, I want to estimate maximum $y$ = $p(y).p(X=x_i|y)$ for $i = 1..n$. ...
2
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1answer
97 views

Quibble with Dawkins's reasoning on the watch-stopping probability on a psychic audience

In Unweaving the Rainbow (page 150) Richard Dawkins mentions the following (famous) reasoning on why it's almost certain that a psychic with a big audience will accurately predict/command rare events ...
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0answers
36 views

Identifying abnormal transactions within large sets of sales data

Exception reporting software is used to identify abnormalities in Point of Sale data. Current software detects abnormalities by comparing an cashier's mean against the store's mean, or the store's ...
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2answers
70 views

Find the median of a function of a normal random variable.

If $X\sim N(\mu,\sigma^2)$ and $Y=e^X$, then what is the median of $Y$? I am pretty sure that $Y$ is also distributed normal. To try to prove it, I attempted both the method of moment generating ...
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1answer
44 views

What is the standard deviation of the sample average (sample size = 16) from a Laplace(0,1) population?

I've been running some simulations, and it seems clear to me that the Sample Mean $\overline{X}$ from a Laplace$(0,1)$ population is distributed normal with mean $0$. But I need to come up with the ...
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1answer
91 views

Prove that the the variance estimator $\widehat{\sigma}^2=MSE/(n-2)$ is biased is the simple linear regression model

This is in scope of the simple linear model. Im trying to prove that $\mathbb{E}\left(\widehat{\sigma}^2\right) = \sigma^2$ for $$\widehat{\sigma}^2 = \frac{1}{n-2}\sum^n_{i=1} ...
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0answers
18 views

pricinpal component analysis recaled mean vector

I am working on Robert Hogg's exercise 3.5.21. I am stuck by question (d) Suppose $\textbf X$ has a multivariate normal distribution with mean 0 and covariance matrix: $\Sigma=\begin{pmatrix} 283 ...
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25 views

Log-likelihood of the normal distribution.

On the attached picture I've highlighted the term which I do not agree with. Is it actually true ? In my calculations I get $$-n(\frac{1}{2}\log(\sqrt{2\pi})+\log(\sigma)),$$ instead. Thank you in ...
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0answers
294 views

Algebra problem involving Q functions

I have the following algebra problem, which is actually the end-part of my bigger research problem. Let $a$, $b$ and $m$ be reals with $a<b$. Also, let $Q(\cdot)$ denote the Gaussian ...
2
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1answer
29 views

Log likelihood function for logistic regression

If the training set S represents are an independent and identically distributed (i.i.d.) sample of a Bernoulli distribution and in logistic regression log likelihood function is given as, ...
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0answers
40 views

Expected run in a run test.

The run test is used to know whether the given data is random or not. How can we derive the formula for the expected run with $n_1$ ups and $n_2$ downs.
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1answer
42 views

Independent, Probabilistic, Event occurs once in a set of 10, Has 10 orientations. Probability?

I have ordered sets. Each index in the set is random and independent. Here is a sample of possible sets: X1 < X2 < X3 < X4 < X5 < X6 < X7 < X8 < X9 < X10: ...
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1answer
54 views

Find the cumulative distribution function.

$f(x)=20 x (1-x)^3$ over $0 \le x \le 1$ and $0$ elsewhere. I know that by definition the cumulative distribution function if $F(x) = P(X \le x) = \int_{-\infty}^x f(t)\,dt$. In this case, I must ...