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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Interpreting Coin Toss Data. Biased or Not?

We have run an experiment in which some good chap has sat down and flipped a coin 100 times. At the end of the 100 flips he has tallied 40 Heads and 60 Tails. Now this seems like something is up with ...
2
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2answers
235 views

Inequality of Pearson correlation coefficient

Let $x_1,\ldots x_n,y_1,\ldots y_n$ be reals and $\bar{x},\bar{y}$ the aritmetic mean of numbers $x_1,\ldots x_n$ and $y_1,\ldots y_n$ respectively. How can I show that $$-1\leq \dfrac{\sum_{i=1}^n ...
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2answers
819 views

Probability of an unprepared student passes a quiz, 3 of 7 must be correct with 4 possible answers each?

A quiz consists of 7 multiple choice questions, each with 4 possible answers. To pass the quiz, it is you must get at least 3 questions correct. An unprepared student can do nothing except guess ...
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1answer
145 views

Let $P(X=1)=0.6$ and $P(X=1,Y>=5)=0.2$. Find $P(X=1,Y<5)$.

I'm thinking with the given information $P(Y\geq5) = P(X=1,Y\geq5)/P(X=1)$, so $P(Y\geq5)=0.33$. Therefore, $P(Y<5) = 1-P(Y\geq5)$, so $P(Y<5)=0.67$. Which means $P(X=1,Y<5) = P(Y<5)P(X=1) ...
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1answer
357 views

Let P(X=3)=0.4 and P(Y=2)=0.5. Find P(X=3,Y=2).

Let $P(X=3)=0.4$ and $P(Y=2)=0.5$. I need to find $P(X=3,Y=2)$. I'm thinking that I ought to just multiply the two probabilities: $P(X=3) \times P(Y=2)$ to get $0.4 \times 0.5 = 0.2$, but is this ...
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2answers
731 views

proving $(cov(X,Y))^2 \leq var(X)var(Y)$ with cauchy schwarz

I've been trying to use the cauchy-schwarz inequality to prove that $(cov(X,Y))^2 \leq var(x)var(Y)$. The cauchy-schwarz inequality can be expressed as follows: if $u$ and $v$ are vectors in an inner ...
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0answers
318 views

circular complex random vectors

Is a vector which components are circular (aka proper) complex random variables also circular complex? Below I summarized my attempt to solve this problem. I think the answer is no, but the ...
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1answer
4k views

Average, standard deviation and min/max values

I'm analzying a computer science paper and just found in the experimental setup the following statement: Average (standard deviation) of number of files per peer: 464 (554) Min - max number of files ...
2
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2answers
215 views

Sum of variances of experiments does not equal variance of joint experiment

Say I make 3 independent experiments and these are the outputs: O/P of $1$st exp : $1,2,3$ O/P of $2$nd exp : $4,5,6$ O/P of $3$rd exp : $7,8,9$ In general ${\rm Var}(A+B+C) = {\rm Var}( A ) ...
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1answer
4k views

Is there a proof of Benford's Law? [duplicate]

As stated by Wikipedia (here): Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed ...
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0answers
18 views

Summation of hypergeometeric series

I was solving a question on hypergeometric, I have been able to solve up to this part and need a simplification. $$ P(x_1)={_{13}}C_{x_1}/_{52}C_5\sum_{x_2 = 0}^{x_2=5} {_{13}}C_{x_2} \times ...
2
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2answers
130 views

flares probability

There are 33 no of flares distributed in 2 hemisphere north and south. the no of flares in north hemisphere is 14 and no of flares in southern hemisphere is 19. then how to find the probability of ...
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1answer
76 views

Probability of winning a game from raw data.

I have a dataset made of couples $(n_i,v_i)$ where $n_i$ denotes the number of times a game has been played, and $v_i$ the number of victories at the $i-$th day. What is the best way to evaluate the ...
5
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3answers
869 views

Expected value of applying the sigmoid function to a normal distribution

Short version: I would like to calculate the expected value if you apply the sigmoid function $\frac{1}{1+e^{-x}}$ to a normal distribution with expected value $\mu$ and standard deviation $\sigma$. ...
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0answers
78 views

Estimating the number of observations from a set of samples

I repeatedly measure a value $S_n$ which is the sum of a set of $n$ hidden inputs. The goal is to identify the number of hidden inputs. All of the hidden inputs are driven by an experimenter ...
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1answer
146 views

how to find the quartiles

I am confused on the method to find the 1st and 3rd quartiles. Suppose given a data set 2,5,6,9,10,11,13,17 with an even number of data. What is the formula to ...
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0answers
53 views

Identification of parameters problem

I always struggle to get the true essence of identification in econometrics. I know that we state that a parameter (say $\hat{\theta}$) can be identified if by simply looking at its (joint) ...
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1answer
113 views

Can I combine probabilities from 2 aspects of a related process?

Consider 2 related aspects of a process for prices in a financial market: time & return. Time Say I've identified an exponential distribution that reasonably models the occurrence of the ...
0
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1answer
876 views

Performing hypothesis test for the difference of population means: paired comparisons

This is a paired comparison question: The answer is: However, this is my thought of using a t test on difference of population means $$t=\frac{583.125 - 415 - ...
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2answers
208 views

Principal Component Analysis

In PCA the first dimension of the basis vector has the highest variance and the last has the least variance. So if we are using PCA just for dimension reduction why cant we find the variance of ...
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2answers
2k views

A tourist in France wants to visit 12 different cities, find the probability.

The Question : A tourist in France wants to visit 12 different cities. If the route is randomly selected, what is the probability that she will visit the cities in alphabetical order? I ...
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4answers
237 views

Maximum Likelihood Estimation and bias

I have a strange (well to me at least) MLE problem. If we let $\{X_i\}_{i=1}^n$ be an i.i.d. sample of a random variable $X$ whose mean is $\mu$ and variance $\sigma^2$. Suppose further that $X_1\sim ...
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2answers
185 views

How to prove the double sample trick inequality $\frac{1}{2} \Pr[|v_1-p| \geq 2 \epsilon] \leq \Pr[|v_1-v_2| \geq \epsilon]$?

$x_i$ is i.i.d random variables with mean $p$. $v_1 = \frac{1}{n}\sum_{i=1}^n{x_i}$, $v_2 = \frac{1}{n}\sum_{i=n+1}^{2n}{x_i}$.Then $\frac{1}{2} \Pr[|v_1-p| \geq 2 \epsilon] \leq \Pr[|v_1-v_2| \geq ...
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2answers
75 views

Finding one value from a mean.

7 boys have a mean height of 1.80m. Peter joins the boys and now the mean height is 1.75m. Find Peter's height.
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1answer
63 views

The CDF of an i.i.d. sample realizations

Is it possible to draw the CDF of the empirical measure $\hat{P}_x$ for a i.i.d. sample realizations where $X_1 = 0.3$,$X_2 = 5$,$X_3 = 1.5$, $X_4 = 3.4$ without knowing the distribution?
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2answers
82 views

Calculate probability that random integer already exists in set of N random integers

I have a set of N random integers between A and B. Assuming that my random number generator is equally likely to return any integer between A and B, how can I calculate the probability that the next ...
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1answer
90 views

Strong Law of Large Numbers Under a Transformation

I have some random variable, $x$, distributed according to a probability density function (pdf), $f\left(x\right)$. The Strong Law of Large Numbers (SLLN) implies that, for an expected value, given ...
2
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1answer
64 views

How do I find the mean with this exponential?

p(y) = c/y^4 I need to find the "mean & variance" of this exponential density function. Some pointers or thoughts that would explain would be most helpful. ...
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2answers
5k views

Calculating missing data points from standard deviation and mean

I have to figure out what to missing data points are from a set of 10 scores. The mean of the 10 observed scores is 20.0 and the standard deviation is 6.0. The observed scores are listed below, with ...
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1answer
483 views

How many degrees of freedom?

I have another question on my stats homework, relating to degrees of freedom. However, I have NO idea how this question even relates to degrees of freedom! I understand degrees of freedom on their ...
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2answers
189 views

How do I account for opposite answers when calculating city block distances?

For a stats class, I have a question in which an occupational therapist uses a checklist about meal preparation. The checklist consisted of five statements to which persons responded using the ...
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1answer
568 views

PMF of a geometric random variable and binomial random variable

We have been discussing pmf. Can someone please show me A. That the pmf for a binomial random variable sums to 1 B. That the pmf for a geometric random variable sum to 1 So I may see the difference ...
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1answer
240 views

Can you please help me find the Moment generating function of the negative binomial when it looks a bit different?

f(x) = $\binom{(r+x)-1}{x-1}p^{r}(1-p)^{x}$ noting 0 < p<1 and r>0 $$\sum_{X}^\infty \binom{(r+x)-1}{r-1}p^{r}(1-p)^{x} =1.$$ x=0,1,2,3......... I think it looks like this: $$\sum_{X}^\infty ...
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2answers
143 views

How to find the mean [E(x)] of this function. Please i know i can find variance after this is shown

Then how do i find the mean of $$\int_0^\infty\frac4{\sqrt{\pi} b^3}x^2e^{-x^2/b^2}dx;$$ if so do how do i do this using the gamma distribution? is there a short cut or do i have to do integration by ...
2
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1answer
86 views

Show that an estimator has a lower variance than another estimator

Let $X $ be observed data. Let $\hat{\theta}(X)$ be an unbiased estimate of $\theta$ and let T be a sufficient statistic for $\theta$. Define the new estimator $\hat\theta^{*}$ of $\theta$, $$ ...
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0answers
72 views

Showing that statistic is unbiased

Let $X $ be observed data. Let $\hat{\theta}(X)$ be an unbiased estimate of $\theta$ and let T be a sucient statistic for $\theta$. Define the new estimator $\hat\theta^{*}$ of $\theta$, $$ ...
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1answer
2k views

Covariance Matrix in Weighted Least Square Estimation

I am new to linear algebra and I have the following doubts: In weighted least square estimation of the system $Ax = b$ we minimize the weighted value of the error $e = b - Ax$ and the best $\hat{x}$ ...
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1answer
236 views

Baseball Probability question

Consider. You throw 3 baseballs at a target. Each baseball has the probability (independent) p= 3/5 of a hit. What's the probability that all three hit? If it is such that a target will be knocked ...
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3answers
399 views

Integration by parts and polar coordinates

Can someone please show me how to integrate $$\int_0^\infty\frac4{\pi b^2}x^2e^{-x^2/b^2}dx\;?$$ please show steps how to integrate this problem. This is what i have so far. $$\frac4{\pi ...
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1answer
45 views

Showing that a set of data with a normal distibution has two distinct groups when you know which point is in which group vs when you don't

Assuming you have a set of data containing two groups, assuming the data would be gaussian distributed, for example height of people, knowing that there are men and women, you can show that the groups ...
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2answers
689 views

monotone functions and pdf's

Could you please show me step by step? Also how does the probability integral transformation come into play? "If the random variable $X$ has pdf $$ f(x)= \begin{cases} \tfrac{1}{2}(x-1)\quad \text{if ...
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2answers
860 views

Least squares estimator of mu

The question is: Assuming that $y_i = \mu + \epsilon_i $,$i = 1,\ldots,n$ with independent and identically distributed errors $\epsilon_i$ such that $E[\epsilon_i] = 0$ and $Var[\epsilon_i] = ...
4
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1answer
192 views

Straight line through data by eye - least squares? [closed]

I heard an interesting fact a while ago about how people draw a line through a cloud of points on a scatter plot. Usually, when calculating lines of best fit, we use the minimal the sum of squares of ...
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1answer
47 views

Average runtime based on X failures

I have a power generator that runs until it fails x times and stops. I want to try to predict how long it will run on average based on known probablities for failures during its operation. The ...
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1answer
124 views

Understanding importance sampling

From Wikipedia Let $X:\Omega\to \mathbb{R}$ be a random variable in some probability space $(\Omega,\mathcal{F},P)$. The basic idea of importance sampling is to change the probability $P$ so ...
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0answers
51 views

Determining The Best Arrangement Of Skills

I have no idea if this question is correct for this forum, but I figured I'd give it a shot. I am trying to figure out the best order to execute skills in a game (I know). Anyway here is the skill ...
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0answers
36 views

statistics- iteration of the alpha parameter in a beta function

I'd like to know whether starting from a Beta distribution and iterating it in the way described below I get a stationary (beta) distribution again. More specifically this is the problem I am facing: ...
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2answers
34k views

What is the equation used to calculate a linear trendline?

In excel it is done automatically but how to manually calculate a linear trendline over a set of points?
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1answer
114 views

Blackjack Basic Strategy

I'm reading Beat the Dealer and I'm confused why you hit when holding a hard twelve against a dealer up card of two or three. You stand on a hard twelve if the dealer shows a 4-6. Can someone ...
2
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2answers
439 views

Exponential Distribution MLE

The lifetime of a type of component has an exponential distribution with rate λ per hour. Ten of these components were tested but the only recorded results were that 3 components had failed within 100 ...