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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Permutations of a sequence of words

I've been given a question in class and I just wanted to confirm the answer ...
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32 views

Is the result of this inquiry statisticaly significant?

So, my family member asked me for help. I was good at math in high-school but I will only have statistics later in collage. Basicly what he asked me for is to - count the p-level - determine if the ...
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41 views

hypothesis testing two sample problem and standard error of mean

In my homework, I was asked to find the standard error of mean for 2 cases and then do the 5% and 1% test for both of them. I know how to work out stand error of mean, but I don't know how to use it ...
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The distribution of the ith order statistic for discrete random variables

Assume $(X_i)_{i=1,...,n}$ are a sequence of real iid random variables with continuous density $p_x$. We know that $$Y:=\sum_{i=1}^n 1\{X_i\leq u\}\sim Bin(n,F_x(u)),$$ since $1\{X_i\leq u\}\sim ...
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Why is the MLE a special case of the minimum contrast estimator?

In my statistics lecture, we had two definitions, namely Let $X_1,\ldots.X_n$ be iid random variables, each with density $p_{\Theta_0}(x)$. Furthermore, let $\varrho$ be a real function such that ...
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171 views

Probability of remaining lifetime using force of mortality

I've been stuck on this question for the past half hour and still have no idea how to solve it... I don't think it's supposed to be very difficult but I'm struggling: There are two independent live ...
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201 views

Find the probability that you and your friend will meet [duplicate]

You agreed to meet a friend of yours at Espresso Royale some time between noon and 1pm. Unfortunately, your cell phone died and you have no way of getting in touch with your friend. Both you and your ...
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98 views

Expected number of coin flips

Assume that when you flip a coin, the probability of getting heads is $1-\alpha$. If you need to flip the coin $N$ times before getting heads, then one can write the expected value of $N$ like so: $$ ...
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4k views

Prove variance in Uniform distribution (continuous)

I read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?
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6answers
4k views

Fewest number of moves to win the game 2048?

I'm trying to figure out the fewest number of moves one could make to win the game 2048. In another thread, someone placed the figure at 520, but I'm wondering if anyone knows how to mathematically ...
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33 views

Odds of getting a certain amount of lottery numbers correct?

If Stacey buys a lottery ticket where she picks 4 numbers, and then 4 balls from a pit of balls labeled 1-20 are drawn (there are only 20 balls, i.e. 1, 2, 3, 4 .. 20) what are the odds she will get 2 ...
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66 views

Switching between two or multiple Poisson processes

Here is the question: Assume that we have $N$ Poisson processes, with arrival rates $\lambda_n, n=1...N$. At the start, we randomly choose, e.g. with equal probability, one Poisson process. Then, ...
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613 views

How many combinations can one make with the following 8 letter word?

Determine the amount of different 3 character combinations you can form with the characters from SEQUENCE. I imagine the solution is $8*7*6$ but EEE and EEE are not invalid, and EEN and EEN are also ...
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111 views

Understanding the Beta-function

I always forget whether the beta function, B$(\alpha, \beta)$, is defined as $\Gamma(\alpha+\beta)/\Gamma(\alpha)\Gamma(\beta)$ or $\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$. Is there an ...
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1answer
74 views

Median of a sequence of random variables.

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables such that almost surely $X_n \rightarrow X$. Given just the information above (i.e. no information about distribution) can one ...
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83 views

A tax on the average and variance

This seems simple but why doesn't this work? "A recent study indicates that the annual cost of maintaining and repairing a car in a town averages 200 with a variance of 260. If a tax of 20% is ...
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56 views

Calculating Variance and Standard Deviation with probability distribution

The age [in years] $X$ of sewing machines to be reconditioned is a random variable with the following probability distribution: $f(x)=(1/972)x(18-x)$ for $0<x<18,$ and $f(x)=0,$ elsewhere. The ...
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532 views

love in an elevator

The sign on the elevator in the Peters Building, which houses the School of Business and Economics at Wilfrid Laurier University, states, "Maximum Capacity 1,140 kilograms (2,500 pounds) or 16 ...
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55 views

Problem related to the exact distribution and the CLT

I'm trying to solve a pretty straight forward problem but i can't find good info on the subjects necessary to solve it so i'm terribly stuck. I'll present it as follows and later try to explain my ...
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1answer
87 views

Randomly choose letters, from the word CHOOSE until both O's have been obtained. Find E(x)

Letters are chosen without replacement. I get it that if I was to choose, lets say the letter C, then my E(x)=(1/6)(1+2+3+4+5+6). Because I have an equal chance to choose the letters for C. But when ...
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424 views

the random heights of north american women

The heights of North American women are normally distributed with a mean of 64 inches and a standard deviation of 2 inches. A random sample of four women is selected. What is the probability that the ...
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646 views

Poisson and exponential distribution problem

The number of planes arriving per day at a small private airport is a random variable having a poison distribution with $\lambda= 28.8$. What's the probability that the time between two such arrivals ...
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Inferring covariance cov[X,Z] from cov[X,Y] and cov[Y,Z] of known distributions

Suppose X, Y and Z are real random variables of known distributions. If one knows the covariance $COV(X,Y)$ and $COV(Y,Z)$, is it possible to infer $COV(X,Z)$?
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135 views

Gumbel distribution

Let $(X_i)_{i \geq 1}$ be a sequence of i.i.d. normal $\mathcal{N}(0,1)$ random variables. Let $M_n = \max_{i=1,\ldots,n} X_i$. Show that $$P[\sqrt{2 \log n} M_n - 2 \log n \leq u ] \rightarrow ...
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93 views

Gaussian Approximation of an intractable distribution

I am currently encountering this problem: I have an intractable distribution and I want to minimize the KL divergence of this distribution and a multivariate gaussian distribution. So we just need ...
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1answer
80 views

Calculate Linear regression segment [duplicate]

I have array of random numbers. How can I calculate linear regression segment? I am interested in finding the exact formula so I be able to use it in my work, please help me finding this formula with ...
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1answer
30 views

Hypothesis testing: how do you call the variable that is being hypothesized about?

The question is easy but it is really hard to find via Google. Say you have the following hypothesis: $H0: \mu = 0$ $Ha: \mu \neq 0 $ Now I know that $ \mu $ is called the population mean. But how ...
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128 views

Minimum of N Chi-square random variables when N is large

I have a problem in numerically evaluating the PDF of $Y=\min(X_1,X_2,\cdots,X_N)$ where $N=\binom{M}{K}$, the binomial coefficient and $X_i$s are iid Chi-square random variables. The CDF of $Y$ is ...
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1answer
65 views

Likelihood function for continuous densities

When doing ML-estimates for discrete distributions, the definition of likelihood makes perfect sense $ \ L(x,\theta) = \Pi_{1:n}\ P(X_i=x_i|\theta)$ Since there is a non-infinitesimal probability ...
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362 views

Curve Fitting a Cyclical Pattern of Data

I'm analyzing phonological characteristics of the 22 letters used in the Hebrew alphabet, and assigned each letter an enumeration to see if they are organized based on place of articulation: ...
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237 views

Algorithm to find best in class of groups with weighting?

I have widgets and a single widget will have attributes of: Name Weight (decimal from 0-1) Group (letter A-F) Price (an integer from 1 - 100) I must pick one ...
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129 views

Bivariate normal distribution problem

Let X be the heigh of the father and Y the height of the son. The two random variables distributed with bivariate normal distribution, as demonstrated by Pearson in 1900. If E [X] = 68 inches and E ...
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81 views

What is this distribution???

Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with $E(X_i) = \mu$ for all $i \in \{1,\ldots, n \}$. Define $ Y_i = \begin{cases} 1 & \mbox{ if } X_i < \mu \\ 0 ...
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44 views

Binomial distribution concept?

The equation for binomial distribution is as follows, $$P(x) = \binom{n}{x}\cdot p^x \cdot (1 - p)^{n-x}$$ My question is, why does it multiply the odds of an event firing with the odds of an event ...
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294 views

Finding probabilities with a Poisson distribution

This is the question that I'm stuck with: An airline knows that overall 3% of passengers do not turn up for flights . The airline decides to adopt a policy of selling more tickets than there are ...
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83 views

Propagation of Error question?

If I have a function $$f(r,V,B)=\frac {2V}{r^2B^2} = 2Vr^{-2}B^{-2}$$ what is the propagation of error? If I use the power rules and multiplication rules described here ...
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42 views

Can multiple events have more than a 100% chance to fire?

For example, if a pitcher has a 50% chance to hit a ball, and during a game he pitches six times, what are the odds he will hit one ball? I would assume the answer is $0.5 * 6$, or $300$%, meaning he ...
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1answer
56 views

Logic behind combinations with repetition?

I've read the stars and bars analogy, but it really doesn't make sense to me. The way I see things, combination with repetition of say 5 of the same color balls in 2 different boxes WITH REPETITION ...
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If $x_1,\cdots,x_n$ are iid normal with mean 0 and variance $\theta$ unknown, find the Jeffry Prior for $\theta$

So I have the following, $L(x|\theta)=-\frac{1}{2}ln(2\pi)-\frac{1}{2}ln(\sigma^{2})-\frac{1}{2\sigma^{2}}x^2$ Then the first derivative is, ...
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19 views

Accuracy Rate with Timing

I have a test that my students take where I am interested in both their accuracy rate as well as the speed in which it took them to complete the test. The results look like this: Bob: 56/66 400s ...
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1answer
2k views

X and Y are two random variables, what is the standard deviation of X - 2Y

I have: E(X) = 10 E(Y) = 12 Var(X) = 4 Var(Y) = 9 covariance = 2 I know that for: ...
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31 views

Logistic regression eye treacting data (need model)

I have two sets of time course data, they are for an eye-tracking study. The data is 20 100ms chunks, one category being percent fixations for canonical sentences, and the other being percent looks ...
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1answer
160 views

Let X1, X2, …, Xn be an iid sample of Bernoulli random variables, Find the likelihood function, MLE,sufficiency

a.Find the likelihood function, L(theta) L(theta)=(x1,x2,x3.....xn|theta)= theta^x(1-theta)^(1-x) ? b. find the MLE MLE is theta =x not sure how to show the sufficiency and how to show the MlE ...
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1answer
34 views

Difference of parameters and arguments when dealing with statistics functions?

In 'classic' math when you have a function like $\sin (\theta)$ or $\cos (\pi)$ is pretty straight forward that both should have an argument and it's very simple to see that in this specific case the ...
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Why does the keyword “distinct” change the solution to much?

I don't understand why the second answer is different from the first. Aren't they the exact same thing? How many ways can we distribute 10 distinct balls into 5 distinct boxes? $5^{10}$ is correct ...
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1answer
47 views

Limiting distribution of $X_n1(|X_n|\le 1-\frac{1}{n})+n1(|X_n|>1-\frac{1}{n})$ if $X_n\sim Unif(-1,1)$ and are iid.

Limiting distribution of $X_n1(|X_n|\le 1-\frac{1}{n})+n1(|X_n|>1-\frac{1}{n})$ if $X_n\sim Unif(-1,1)$ and are iid. From looking at the term, if $n$ goes to infinity, then $Y_n$ would be $X_n$ so ...
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2answers
440 views

What is the probability that $x_1+x_2+…+x_n \le n$?

Given that $X_1, X_2...$ are mutually independent random variables. For each $i$ with $1\le i \le n$ the variable $X_i$ is equal to either $0$ or $n+1$ $E(X_i)$ = $1$ also.. if $X_i$ is equal to ...
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1answer
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Differentiate $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+…+z^{i-1}}{i}$ twice to calculate the variance of involutions.

Use the Probability Generating Function for Involutions: $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+...+z^{i-1}}{i}$ To Calculate the Variance of Involutions where: $Variance \space X_n = ...
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Distribution of the Objective Value and the Variables in an Optimization Program

For random variables $X$ and $Y$, where $X\sim f(X;\theta)$ ($X$ is drawn from some distribution with pdf $f$ which is parametrized by $\theta$ ), $Y=g(X)$; we know that we can find the pdf of $Y$ if ...
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5k views

How do I find if the probability of the sample proportion is greater than something?

I have this problem and I have no clue how to solve it. In 2012, 31% of the adult population in the US had earned a bachelor’s degree or higher. One hundred people are randomly sampled from the ...