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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Showing that the sample variance for an SRS is a biased estimator of the population variance?

EDIT: I suspect I may be going about this all wrong, so maybe disregard this. So, I can get this far on my own: $E(\hat{\sigma}^2) = E(\frac{1}{n - 1}\sum_{i = 1}^n (X_i - \mu)^2) = \frac{1}{n - ...
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2answers
57 views

Question about Bernoulli Distribution calculation

can sombody explain the above calculation in the red circle marked with "why?"? I am studying MLE with Bernoulli Distribution, and in the middle of a video clip, the lecturer says $ 1\over{n} ...
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1answer
86 views

find a function that is an unbiased estimator

Let $X_1,X_2,...,X_n$ be a random sample from the distribution $$f(x;p)=p(1-p)^{x-1}$$ where $x=1,2,...$ and $0<p<1$. I know that the sufficient statistic is $Y=\sum X_i$. Now I have to find ...
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1answer
41 views

Getting the right P value

i want to check if there is a significant difference between in gasoline consumption between gas-1 and gas-2: here are some observations from gas-1 ...
2
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1answer
65 views

Probability Question I can't get around.

This is the question from my assignment, which I can't get around. Suppose that a water distribution system is composed of a number of independent pipes. At temperatures below 0 deg C, the pipes ...
2
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1answer
291 views

How to calculate the Lambda of Poisson distribution from mean of inter arrival time?

I have inter arrival time into a system mean equal to$ 0.45.$ Does $\lambda = \frac{1}{0.45}$ if I need to select Poisson as an arrival distribution?
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4answers
592 views

What happens if I toss a coin with decreasing probability to get a head?

Yesterday night, while I was trying to sleep, I found myself stuck with a simple statistics problem. Let's imagine we have a "magical coin", which is completely identical to a normal coin but for a ...
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1answer
144 views

suppose x follows a distribution with density function: f(x)=C*abs(x-2), 0 =<x=<3; f(x)=0 otherwise.

suppose x follows a distribution with density function: f(x)=C*|x-2|, 0 <=x<=3; f(x)=0 otherwise. find the cumulative distribution function of F(x) for 2<=x<=3 Find the Median of the ...
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1answer
69 views

Given $N$ coins, find a coin with minimal bias based on $N$ samples

General description: Given $N$ coins $Z_1,...,Z_N$ (Bernoulli RVs), where the $i$-th coin has probability $p_i$ for "Head", I'm trying to find $\min\limits _{i\in[N]}p_{i}$. I'm interested in a ...
2
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1answer
43 views

Max Word Size as a Function of Number of Words

I want to describe the relationship between largest word length, l, and the number of words in a set, n. Example: For the set ...
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3answers
1k views

what is the probability of a couple who has four girls and is trying again for a boy. what is the probability that the next kid will be a girl?

A couple has four kids already, all girls. the couple would like to have a son and would like to give it another try. what is the probability that the next kid will be a girl?
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1answer
123 views

How to appropriately normalize financial data?

I am evaluating a normalization of financial data. Two colleagues feel certain of a certain approach, and I am befuddled. The approach shapes the results how they want (i.e., brings possibly extreme ...
3
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1answer
96 views

Repeated sampling with replacement, increasing probability

I would appreciate help with the following problem, since I can't quite figure out the effect an increasing number of trials has on probability: Suppose a bin has white marbles and black marbles. Say ...
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2answers
81 views

Can this be solved?

Suppose that 47% of all Americans have flown in an airplane at least once and that 28% of all Americans have ridden on a train at least once. What is the probability that a randomly selected American ...
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1answer
113 views

delta method question

Let $H:\mathbb{R}^k\to \mathbb{R}^k$ be measurable and differentiable at $x_0$, i.e. $$H(x) = H(x_0) + L(x-x_0) + o(x-x_0)$$ near $x_0$. Suppose $\{X_n\}$ and $X$ are random vectors in $\mathbb{R}^k$ ...
2
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1answer
121 views

Calcute a chance that whole rectangle lays inside of the circle

We are given circle with radius 1. Point P lays somewhere on that circle picked from the uniform distribution. $\{P_x^2+P_y^2 = 1\}$ Point Q as well was randomly picked from the uniform distribution ...
2
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0answers
126 views

Best closed convex surface fitting N points in 3D

First. It's easier to understand the problem by describing the application where it arises from. We have a convex body $B$ in $\mathbb{R}^{3}$ and measure points on its surface. The measurements are ...
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1answer
66 views

how do i find the formula?

A outdoor light bulb has an expected (mean) life of 8,000 hours with a standard deviation of 250 hours. How many bulbs in a batch of 500 can be expected to last no longer than 7500 hours?
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1answer
31 views

What is the standard deviation of the score?

A card is drawn from a deck of 52. The score equal to its rank unless it is a court card (Jack, Queen or King) with a score of 10, otherwise equal to its rank and Ace counts as one. What is the ...
2
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1answer
139 views

How to rank a separate population using elo points/system

Background: I have a website where students vote on the attractiveness of their peers: they are presented with two images, and they must pick one (the "winner")- then the elo score for each is ...
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2answers
71 views

calculate a 90% confidence interval

There's a report saying that $67\%$ of teachers surveyed think that computers are now essential tools in the classroom. Suppose that this information was based on a random sample of $n=200$ teachers. ...
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1answer
165 views

Central Limit Theorem - Distribution Function Converges to Standard Normal

Suppose that $X_1, X_2, ..., X_n$ and $Y_1, Y_2, ..., Y_n$ are independent random samples from populations with means $\mu_1$ and $\mu_2$ and variances $\sigma_1^2$ and $\sigma_2^2$, respectively. ...
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1answer
51 views

Unbiased estimate $\lambda^2$

Given a Poisson distribution I want to figure out whether $d:(x_1,...,x_n) \mapsto x_1^2$ and $d':(x_1,...,x_n) \mapsto x_1x_2$ are unbiased estimations for $\lambda^2$ ? I mean it would sound ...
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1answer
3k views

Chance of randomly guessing 21 questions right out of 50 with 4 multiple choice.

Lets say a person decided to randomly fill in a scantron of 50 questions with 4 choices each. After submitting it to be graded, the result was 42% correct. How would we figure out the probability of ...
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4answers
134 views

Probability & Statistics: Random variables

I have a problem similar to the well-known "Coupon Collector Problem." A box of a certain brand of cereal comes with a special toy. There are 10 different toys in all. How many packs you will need ...
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1answer
236 views

Difference between two sample proportions

Just as the difference between two sample means is normally distributed for large samples, so is the difference between two sample proportions. That is, if Y1 and Y2 are independent binomial random ...
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1answer
155 views

Statistics - Approximating Poisson Distribution

Y, the number of accidents per year at a given intersection, is assumed to have a Poisson distribution. Over the past few years, an average of 36 accidents per year have occurred at this intersection. ...
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1answer
1k views

Complete Statistic: Uniform distribution

Take a random sample $X_1, X_2,\ldots X_n$ from the distribution $f(x;\theta)=1/\theta$ for $0\le x\le \theta$. I need to show that $Y=\max(X_1,X_2,...,X_n)$ is complete. Now, I know I should ...
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0answers
17 views

Any way to simplify this expression?

So I have a vector of asset allocation weights given by $x \in R^4$ and a covariance matrix of the asset returns $\Sigma \in R^{4,4}$. I know by the spectral theorem, $\Sigma = V DV^{-1}$ and the ...
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1answer
298 views

Proof that a median minimizes 1-norm. [duplicate]

I was wondering whether there is an easy way to show the following: We have a data set $x_1,...,x_n$ and $m$ is a median if for at least half of the n data points we have that $x_i \le m$ and for ...
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1answer
159 views

Dice Probability of rolling at least one four?

One dice is rolled three times. What is the probability of getting at least one four? I've been getting stuck on what to do next I know that it could be one four, two fours, or all three rolls could ...
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3answers
34 views

Coin and steps -Probabilty and Statistics

Turn a coin and if it falls heads move three places to the right otherwise move 2 places left. After 20 times you turn the coin, in what positions might you be and what is the probability to be in ...
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1answer
29 views

The minimal sufficient statistic is not unique, but the minimal sufficient partition is unique, what does this mean?

Can anyone help to illustrate the statement with some examples?
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1answer
24 views

The proof to the statistic estimate

Suppose we have $$r=\frac{a_1a_2 \cdots a_m}{b_1b_2 \cdots b_n}$$ Relative error of each of $ai$ and $b_i$ is roughly the same and equals $\delta$. There is a theorem which says: ...
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1answer
30 views

Error of function with errors on arguments and size of arguments

EDIT: my question is not how to derive the formula below (I think the derivation is more or less what I guessed, like the answer below supports), but whether it can be made valid for the case where ...
2
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1answer
82 views

Let $X$ be symmetrically distributed about 0. Show that $X$ and -$X$ are identically distributed

Hello Mathematicians! The definition of being symmetrically distributed about zero means that the PDF (Continuous) or PMF (Discrete) $f(x) = f(-x)$. And the definition of being identically ...
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1answer
283 views

Determine the distribution underlying several given scores and their percentiles

I am trying to build a FICO score calculator that estimates one's FICO score given one's percentile from another credit report. FICO score data is kept fairly secret, but the following information is ...
2
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1answer
98 views

Is my understanding of the Central Limit Theorem correct?

Have I got this correct - Say we have a population. We take a random sample of size $n$ from this population. I.e. we form a sample $S$ based on random variables $X_1, X_2, ..., X_n$ taken from this ...
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1answer
369 views

cumulative standard normal distribution formula

I need to calculate a P-value (for significance checking) out of the Z value, mean(0), standarddeviation(1), normal distrubution being cummulative. Is there a function in PHP that could do that? ...
2
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0answers
52 views

What's the proper graph representation for a category prices trends in time?

As I'm not a mathematician I thought I'd ask here for advice how to approach something I'm working on (probably a basic question for a lot of you guys, but it was a subject of a debate at my work ...
2
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1answer
123 views

Confidence level of random sample from continuous distribution

Let $X_1,X_2,\cdots ,X_n$ be a random sample from a continuous distribution with median $\mu$. If $[X_{min}, X_{max}]$ is used as a confidence interval for $\mu$, what is its confidence level? What is ...
3
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2answers
54 views

Minimum of variance when sample is unbiased?

Show that if an estimator $\hat\mu=a_1X_1 +a_2X_2 +\cdots+a_nX_n$, where $a_1, a_2,\ldots,a_n$ are constants, is unbiased, then its variance is minimum when $a_1=a_2=\cdots=a_n=\frac{1}{n} ...
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0answers
53 views

Metric for precision of a decimal number

I am working on cleansing a cities database and trying to increase the precision of the latitudes and longitudes of these cities. I am comparing what I have against an outside datasource and want to ...
0
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1answer
36 views

How can we evaluate the graph by CDF?

Could you please help me to evaluate the graph correctly? Basically, the graph shows 3 lines which are early adopters of a convention, rest of the adopters and the all users. The x-axis is the number ...
0
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1answer
88 views

Absolute values of a linear combination of three random variables

From my stat homework: with $X \sim N(3.2, 6.5)$, $Y \sim N(-2.1,3.5)$, $Z \sim N(12.0,7.5)$ (all are independent random variables) find probability that: $$ |X + 6Y + Z| \geq 2 $$ I have $(X + 6Y + ...
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1answer
72 views

A simple binomial distribution problem why is my textbook saying I am getting the answer wrong.

Hello everybody I am doing some homework problems in preparation for my upcoming exam. The name of the course is Statistics for Engineers. Anyhow here is the problem description: Let $X$ denote a ...
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1answer
2k views

Find an unbiased estimator function (Poisson)?

I think it is pretty easy to find an unbiased estimator for a regular distribution, whether it be Poisson or Gamma or something else. For example, the unbiased estimator (Poisson from a random sample ...
2
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1answer
51 views

On track Prerequisite for Statistics and Probability

I do not really have a solid mathematical background because of the range of courses i had back in high school/university that wasn't really scientific oriented. Presently i am doing an MSc in ...
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1answer
53 views

Binomial probability formula

The binomial probability formula goes like this, $$ f\left(x\mid p\right) = p^{x_i}\left(1-p\right)^{1-x_i} $$ But I wonder why the success probability and failure probability are multiplied? Can ...
0
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1answer
34 views

Given 50 IID normals, find the exact SE for the estimate of $\sigma^2$?

Given 50 I.I.D Normal distributions random variables $X_i$, the Maximum Likelihood estimator for $\sigma^2$ is $\hat{\sigma}^2$, as proven in my lecture notes. Find the EXACT SE. My Attempt: ...