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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2answers
73 views

Is the maximum-likelihood estimation notation formally correct?

I just saw from the Wikipedia's entry on Maximum likelihood, http://en.wikipedia.org/wiki/Maximum_likelihood , the formula $\mathcal{L}(\theta\,|\,x_1,\ldots,x_n) = f(x_1,x_2,\ldots,x_n\;|\;\theta) = ...
4
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1answer
142 views

Philosophy of Statistics (Likelihood Function)

Last week during statistics class, my professor asked us a few basic questions about statistics. We could answer most of them except these three questions that we could not provide him good answers. ...
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3answers
4k views

Odds of two dice roll

If we roll two dice, what are the odds that we roll one five OR the sum of the rolled dice equals an odd number? The odds of rolling one five from two dice rolls is $\frac{1}{36}$. The odds of ...
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0answers
79 views

Probability and Statistics: Bowling Night Poker

Let me build this up a bit. There's five people on my bowling team, including myself. None of us are especially good at bowling; our averages range from about 120 to about 155 (Mine is the 155, ...
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3answers
101 views

Why is the statement false: If a test is rejected at significance level α, the probability that the null hypothesis is true equals α? [closed]

Since the answer is false, so it just means the probability that null hypothesis is false equals α. But α is defined as the probability that null hypothesis is rejected when it is actually correct. I ...
1
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2answers
64 views

Why does variance divide by $n-1$? [duplicate]

The variance is: $$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$ I read that $n-1$ is used instead of just $n$ when we are measuring the variance of a sample taken from a bigger population. I don't ...
0
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2answers
34 views

Minimum amount of data storage required for keeping track of variance? [duplicate]

Let's say I have n elements to begin with, and over time more elements are added in. What's the minimum amount of data I have to store in order to be able to continuously update the variance, and how ...
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7answers
149 views

Why is variance squared?

The mean absolute deviation is: $$\dfrac{\sum_{i=1}^{n}|x_i-\bar x|}{n}$$ The variance is: $$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$ So the mean deviation and the variance are ...
1
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1answer
119 views

order statistics question 1

Let $Y1,Y2,...,Yn$ be independent, uniformly distributed random variables on the interval $[0, θ]$, $Y(k)$ the kth-order statistic, where k is an integer between $1$ and $n$. Find $E(Y(k)- Y(k-1))$, ...
1
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3answers
60 views

question about Mean absolute deviation formula

$$\frac{\sum_{i=1}^{n}(x_i-\bar x)}{n}$$ This method will not work for calculating the mean deviation. Instead we have: $$\frac{\sum_{i=1}^{n}|x_i-\bar x|}{n}.$$ I'm not quite understanding why ...
1
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2answers
188 views

What is the meaning of Common Support here

I am reading a notes in statistical inference, and I am constantly being confused about the term 'common support', i hardly find any definition of this,here is an example, 'Suppose S is a space of ...
1
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1answer
64 views

When using Bayes Rule, what are the rules for flipping the conditions and the event of interest?

Here is Bayes Rule: $$P(A\mid B) = \frac{P(B\mid A) P(A)}{P(B)}$$ This paper (http://www.cogsci.northwestern.edu/Bayes/Sivia_1996.pdf) uses Bayes rule on page 21 in the context of model selection ...
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0answers
86 views

Choosing between increasing sample size and testing for failures

I have a large set of data that shows things (let's call them cars) have performed over the years against various different tests (say, crash test, braking, gas mileage, etc.). What I'm doing is ...
14
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3answers
8k views

The Best Strategy and Highest Possible Score for the “Threes!” Game.

[There's still the strategy to go. A suitably robust argument that establishes what is statistically the best strategy will be accepted.] Here's my description of the game: There's a $4\times 4$ ...
2
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2answers
142 views

derivation of simple linear regression parameters

I know there are some proof in the internet, but I attempted to proove the formulas for the intercept and the slope in simple linear regression using Least squares, some algebra, and partial ...
1
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2answers
65 views

Product & Ratio's of 2 Random Variables

I'm interested to know whether it's the case that for random variables $X$ and $Y$ whether or not the ratio of $X$ and $Y$ can be computed as the product of $X$ and $1/Y$. That is, Is $\frac{X}{Y} ...
0
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0answers
50 views

Standard Deviation ( Sort of ) Problem

I have a bunch of particles spread over a 2D area. I plan to fit a a distribution to describe the spread. To start off : I have about 100 particles arranged in a circular pattern in a "unit" ...
1
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1answer
48 views

Finding the cdf of a distribution

Two people wish to explore Asia, where both tourists have a lifetime that is exponentially distributed with mean $\theta$. $T$ is the length of time that Asia is being explored by at least one of the ...
1
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2answers
601 views

A Flat Tire Excuse

I have this multi-part question on an assignment that I don't understand. Hopefully someone can help. There's a story that 4 students missed their final and asked their professor for a make-up exam ...
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1answer
35 views

Calculating number on normal distribution curve

Can someone please let me know if I have this question correct: ...
0
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1answer
308 views

how many tickets should i buy in this raffle?

There is a raffle and there are 500 raffle tickets for sale (assume they all get sold) In the raffle there are 10 prizes to be won. There is one prize I particulary want to win (I don't bother about ...
-1
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2answers
57 views

Poisson Distriburion

I have a problem that follows Poisson distribution with $\nu=10/6$ and $t=1$. So the formula has this form: $$\mathbb{P}(x)=\frac{(10/6)^x}{x!} e^{-10/6}.$$ I have to find the $x$ assuming that ...
2
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0answers
36 views

Smallest set of Liner equations, which exactly fit a set of points

I have a set of 2-d points,(it can be of any arbitrary dimension n). I want to find the minimum set of straight lines(linear equations) which exactly passes through the given 2-d points (unlike ...
0
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1answer
35 views

Finding pmf of sequential experiment

I have $X,Y\sim Poisson(\lambda)$ and if I observe $X=0$ first, no $Y$ is observed. If $X>0$, then we observe a $Y$. I want the pmf of $T=0$ if $X=0$ and $T=1+X+Y$ if $X>0$. I know the joint ...
1
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1answer
48 views

How can the Wilson Confidence Interval be adapted for more than 2 outcomes?

In this link, the Wilson Score Interval is used to calculate the interval of a discrete distribution in which possible outcomes are 1 star, ...
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2answers
532 views

Proving a sample mean converges in probability to the true mean

I have an i.i.d. sequence of observations $\{x_1, x_2,\ldots, x_n\}$ and I need to prove that $\frac{1}{n}\sum x_i$ converges in probability the "true mean", that it is consistent estimate of the ...
2
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0answers
89 views

Model selection: geometric mean of the standard deviation.

I have two models that represent a physical process. To determine which model is the best, I make some experiments and compare measured data with data predicted by each of the models. The model with ...
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3answers
2k views

Mean or median when data interpretation

When doing results treatment, sometimes the mean values agree more or less with the median, but sometimes not. Thus using one or other value may change the final results. Then, is there any criteria ...
0
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2answers
90 views

Statistics - scaling sets of data - mean & S.d

A set of numbers has a mean of 22 and a standard deviation of 6. If 3 is added to each number of the set, and each resulting number is then doubled, find the mean(50) and standard deviation(12) of the ...
0
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1answer
28 views

Functions of random variables and joint pdf's

Random variables X,Y have joint pdf as f_X,Y ^ (x,y) = 24 x.y and x,y>0 x+y<1 find the marginal pdf's of U=X+Y and V = X/Y i tried to solve it and got that: g_V,U (v,u) = 24 . v . u^3 / (1+v)^4 ...
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1answer
61 views

Where are they getting this number from?

Here's the question that I'm having a problem with: ...
1
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1answer
85 views

Determine the sample size, statistics

My professor sent us a homework question that had me puzzled. The homework consisted of four questions that were related and fifth one that is totally independent of the first four. The first four ...
2
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0answers
68 views

Help understanding interaction and effect.

I am reading a research paper in which the authors designed three bots and wanted to determine what bot was most human-like (link is at the bottom). To do this they had subjects play against each bot ...
0
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2answers
557 views

the profit of pizza

The number of pizzas delivered to university students each month is a random variable with the following probability distribution $x=\{0,1,2,3\}$ and $$P(X=x)=\{.1,.3,.4,.2\}$$ respectively. If the ...
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0answers
82 views

Is $Z_n$ a Martingale with mean $1$?

Consider a sequence of independent tosses of a coin, and let $P_h$ be the probability of a head on any toss. Let $A$ be the hypothesis that $P_h = a$, and let $B$ be the hypothesis that $P_h = b$. Let ...
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0answers
110 views

Multivariate convergence in distribution

Assume $X_i$ are iid with mean 0 and variance $\sigma^2$ and $E(X^3_i) =0$. Define $\bar{X}$ and $S^2 = \frac{\sum(X_i^2)}{n}-\bar{X}^2$ Prove that convergence in Distribution of $$ \sqrt(n) ...
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0answers
54 views

Expected Sum of Weights after Drawing Without Replacement

We have an urn containing $k$ balls where for all $i:1\le i\le k$, the ball $b_i$ has the size $s_i$ that determines its probability to be drawn. For instance, a ball $b_i$ with size $s_i=3$ is ...
1
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1answer
85 views

Find the limiting distribution of the following random variable

Let $X_1,X_2,...$ Be independent random variables with common density: $$f_X(x)=\alpha x^{-(\alpha+1)}. x>1$$ Where $\alpha>0$. Define a new sequence of random variables: ...
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0answers
596 views

How to interpret coefficients in polynomial regression?

I am working on my thesis (study) about poverty incidence rate and its socio-economic factors using second-order polynomial regression without interaction. The final model in my study is ...
1
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1answer
54 views

CLT approximation

Let $X_1,\ldots,X_{735},Y_1,\ldots,Y_{880}$ be independent random variables such that $P(X_i=0)=\frac{3}{7}$, $P(X_i=1)=\frac{4}{7}$ and $P(Y_i=0)=P(Y_i=1)=\frac{1}{2}$. Find $P(\sum_{i=1}^{735} X_i ...
0
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2answers
415 views

craps probability question

The following exercise is best solved with a computer. The probability of winning a game of craps (a dice-throwing game played in casinos) is 244/495. a. What is the probability of winning 5 or more ...
2
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0answers
32 views

Prove the relatinship between Beta Distribution and Bionomial Distribution

Prove that $$ \sum_{k=0}^{x}{n\choose k} p^k (a-p)^{n-k} = (n-x){n\choose x}\int_{0}^{1-p}t^{n-x-1}(1-t)^{x}dt $$ (Hint : Integrate by parts or differentiate both sides with respect to $p$) From the ...
0
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2answers
723 views

How to increase mean of set of numbers

Lets say: $X = \{x_1, x_2, x_3, ... \} $ be a set of Real numbers in range $(R_1, R_2)$ and $m =$ mean of $x$ If I have to increase mean of set $X$ by $3$, each number in the set has to be increase ...
-2
votes
1answer
47 views

Show that $E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $ for a martingale with $Z_0=0$

I was just wondering, if we let $(Z_n)_{n\geq 0}$be a martingale with $Z_0=0$, is it true then $$ E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $$ Please let me know and if it is true, can someone show ...
0
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2answers
104 views

Probabilities and Binomial Distributions

Suppose $X$ is a binomial random variable with $n = 25$ and $p = 0.7$. Use Excel or Minitab to find the following. Please give the answers to five decimal places. $$P(X = 16) = ?$$ I used the excel ...
1
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1answer
46 views

What is the rationale behind ROC curves?

I am not sure how ROC curves work. I see that the X-Axis is the false positive rate while the Y axis is the true positive rate. 1) I don't understand how for a given statistical learning model, you ...
1
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1answer
398 views

Measuring degrees of randomness

Imagine, for simplicity's sake, that we have a set of numbers, each equal to either 0 or 1. Let's call each a bit. Rationally, if the set is completely random, and reasonably large, the probability ...
0
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1answer
116 views

variance of minimum of squared exponential random variable

Given $Y_1 $ to $Y_n$ are exponential r.v's with mean $\theta$ find $\operatorname{var}[\min(Y^2 )]$ with the help of gamma distribution. attempt: $\min(Y) $ is exponential with $(\text{mean} = ...
1
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1answer
121 views

Bayesian Probability Question - Parameter Estimation

I would like help on the following question and I will show my work. Here is the question in my notes and I will follow up with my work: Q: Suppose a forest is segmented into strips, referred to as ...
2
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2answers
340 views

Probability of always rolling 6 on a dice

Suppose I roll a six-sided die $10$ times and each time it shows a $6$. What is the probability of the next roll coming up $6$? You might say $1/6$. But it was never declared to be a fair die. In ...