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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There are 5 servers. Each server has 1% downtime. What's the probability that at at least three servers are down?

There are 5 servers. Each server has $1$% downtime. What's the probability that at at least three servers are down? My reasoning is the following: A) There is $(1-0.01)^5$ probability that $5$ from ...
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31 views

( X ~ $B(n_1, p) $, Y ~ $B(n_2,p)$, (X,Y) independent ) $\implies Z = X + Y$ ~$ B(n_1 + n_2, p)$

Hello I have some exercise from probability and statistics, but I don't know what they are really asking about: Random variable X has Binominal distribution $B(n_1, p)$. Random variable Y has also ...
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39 views

Bounded function of geometric random variable

if X~ Geometric(p), with q=1-p, then show that for any bounded function f with f(0)=0, we have E(f(x)-qf(x)+1)]=0. Our professor asked us to try solving this problem as a good practice but I have no ...
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91 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
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30 views

Confidence interval for a binomially distributed observation with few trials?

If there are few trials and you want to get the confidence interval of a binomially distributed observation, is it still okay to use the normal approximation interval, or is that only accurate for a ...
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55 views

Derive a model for the variances $\sigma_i^2$ for which $b_1$ is the best linear unbiased estimator (BLUE) of $\beta$

Consider the model $y_i=\beta x_i + \epsilon_i$ (without a constant term and with $k=1$), where $\mathbb{E}[\epsilon_i]=0, \mathbb{E}[\epsilon_i \epsilon_j]=0, \forall i \neq j$, and ...
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1answer
45 views

Determining total number of tickets sold

Let's say there is a lottery where ticket numbers 58,145,350, and 400 won. Is there a way to determine the most likely number of tickets sold? We are assuming non-replacement and fair draws.
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158 views

normalizing a non gaussian distribution

I have a set of empirical data that I have collected from different people with several attributes. For privacy reasons I will refer to these attributes as a,b,c,d,... etc. Since the data was ...
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1answer
230 views

Clarify my understanding for central limit theorem from a statement

Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student ...
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1answer
53 views

What type of distriburtion?

A comparative study of responses to a questionnaire addressed to consumers, showed that the lifetime of televisions usually distributed with an average of 8.2 years and a standard deviation of 1.3 ...
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3k views

How to find (and plot) a probability distribution function?

I'm working on my biometrics course, and I have to plot a pdf (I think it means probability density / distribution function). Here is a sample pdf graph : Introduction to Biometrics page 5 , figure ...
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62 views

Statistics & Probability -Distribution problem

10% of the chocolate bars that are produced in a factory have unacceptable shape. In a sample of 1000 chocolate bars find the probability that the number of unacceptable shapes is (A) less than 80 ...
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57 views

Sum of Random Distributions/ Unusual Results

$$X \sim N(\mu_1,\sigma_1^2)$$ $$Y \sim N(\mu_2,\sigma_2^2)$$ then $$X+Y \sim N(0,\sigma_1^2+\sigma_2^2)$$ One way, I tested this to be true is in excel, I used the ...
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2answers
40 views

Find a minimum of a variance

Given $Y_1$ and $Y_2$, linearly independent. $Y_1$ has mean $\theta$ and variance $\sigma_1^2$, and $Y_2$ has mean $\theta$ and variance $\sigma_2^2$ Given $Y_3 = a Y_1 + (1-a) Y_2$ So then by ...
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1answer
36 views

How does effect size affect alpha?

As the effect size increases the power of a hypothesis test also increases, but what happens to alpha?
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5answers
109 views

Why square the result of $x_1 - \bar{x}$ in the standard deviation? [duplicate]

I don't understand the necessity of square the result of $x_1 - \bar{x}$ in $$\sqrt{\frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N-1}}$$. In fact I don't understand even why is $N - 1$ on the denominator ...
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1answer
323 views

I have trouble understanding the proof of the Wold decomposition theorem

I'm trying to understand the proof of the Wold decomposition theorem in [1, p.187]. I find a few things about it very irritating. The theorem states: Theorem 5.7.1 (The Wold Decomposition). Let ...
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1answer
68 views

Probabilty & Statistics Problem

The number of cracks which are present in a part of an international road has an average value of 2 cracks per kilometer. 1)What is the probability that there are no cracks in a section of road ...
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1answer
163 views

Dummy variable in the probability generating function

I'm struggling to understand what the purpose of the dummy variable $t$ in the probability generating function is? I know it takes a value between 0 and 1, and have heard it described as a 'relative ...
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2answers
66 views

Proper name for the generalized binomial distribution with two trials?

Two Bernoulli trials are performed, first with success probability $p_1$ and second with success probability $p_2 \not=p_1$. The resulting distribution for the number of positive outcomes is ...
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0answers
43 views

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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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 ...
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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 ...
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1answer
281 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
577 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
141 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
68 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 ...
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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 ...
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1answer
95 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 $\mathbb{R}^k$ ...
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1answer
118 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 ...
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0answers
125 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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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 ...
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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
157 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
2k 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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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
235 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
141 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
972 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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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
292 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 ...