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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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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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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69 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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621 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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115 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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78 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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87 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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543 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 Persons....
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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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93 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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430 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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658 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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137 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 e^{-e^...
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96 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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82 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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129 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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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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374 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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2answers
243 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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132 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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1answer
309 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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84 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 http://www.fas.harvard.edu/~...
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43 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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57 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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17 views

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, $-\frac{1}{2\sigma^{2}}+\frac{x^{2}}{(\sigma^{2})^{2}}$...
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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: Var(X + Y) = Var(X) + Var(Y) + 2covariance ...
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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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163 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 as ...
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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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529 views

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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442 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
94 views

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 = P_{x_n}''(1)...
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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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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 ...
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118 views

Maxima CAS , Statistic Formula with Maxima

I am working on my graduation project which study sport players' performance. I have collected the data and finished all the surveys but then I am stuck at using the following formula to calculate ...
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54 views

Statistics Similar Number in sets

I'am trying to determine what is the likelihood having the same sequence of numbers in a given array. I have an array of 2 rows and n columns (n is known), I then decided a range (1 to x) let's say 6 ...
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35 views

Probability that certain drawn pieces of paper having the same colour are in the same group

I have the following question: I have 12 players and wrote their names on 12 pieces of paper. I put the 12 pieces of paper in a box, but 2 pieces of paper were green-coloured, and the rest was red-...
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2answers
156 views

Likelihood Function for the Uniform Density $(\theta, \theta+1)$

Let the random variable X have a uniform density given by $f(x;\theta)$~$R(\theta,\theta+1)$ What is the maximum likelihood function according to the samples $X_1,\ldots,X_n$? The question is much ...
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624 views

probability of getting particular cards in a hand (cards)

I was given the question in class: "What is the probability of getting a hand with 1 heart, 2 diamonds, 2 clubs?" and "What is the probability of getting a hand with at least 3 queens" for the ...
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2answers
1k views

Permutations of a letter sequence

I have been given a question about (i think) permutations, it asks to find how many x letter sequences can be made out of a word. I am getting confused because it first asks how many different ...
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1answer
36 views

Permutations of a word

I was asked in class how many unique 5 letter combinations I can make out of the word "maths" i know that you can use the permutations formula to work out the maximum amount of combinations of this ...