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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$E[\hat{\theta}_{MME}] = E[\frac{1- 2\overline{y}}{\overline{y}-1}] = \int_0^1 \frac{1- 2\overline{y}}{\overline{y}-1}(\theta+1)y^\theta dy$..?

Let $Y_1, Y_2,\dots , Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
29 views

Calculating probability of a dice with different numbers [duplicate]

I was given a problem in class: ...
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1answer
660 views

Unbiased estimators in an exponential distribution

We have $Y_{1}, Y_{2}, Y_{3}$ a random sample from an exponential distribution with the density function $ f(y) = \left\{ \begin{array}{ll} (1/\theta)\mathrm{e}^{-y/\theta} & y \gt 0 \\ 0 ...
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2answers
60 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
51 views

poisson and discrete distribution

Business failures are due to three mutually exclusive risks: market risk, credit risk, and operation risk, which account for 20%, 30%, and 50%, respectively, of all business failures. Suppose the ...
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1answer
49 views

Chances of winning a raffle when winning tickets are returned to bucket each time.

first time posting here - I like the site. I have a raffle odds question. I'm doing a raffle where I give away 365 prizes. Every winning ticket is returned to the barrel after each drawing (and ...
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1answer
45 views

Probability question! with and without replacements.

Urn contains $10$ blue balls, $5$ red balls, $5$ green balls. When $9$ balls are selected, what is the probability of $7$ balls being blue? i) with replacement ii) without ...
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1answer
26 views

Calculation of ${\rm E}[e^X]$ where $X$ follows a uniform distribution over $(1,2)$

To compute ${\rm E}[e^X]$, in which $X$ has uniform $U(1,2)$ distribution. Here $f(x)=1$ for $x\in (1,2)$. The formula is $\int_0^1 e^x \,\mathrm dx$. Is the answer $e^2 - e^1$?
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61 views

Probability Question

The credit department of a clothing shop reported that 30% of their sales are through cash/check, 30% are through credit card and 40% are through debit card. 20% of the cash/check purchases, 90% of ...
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1answer
26 views

Expected value and variance of n independent links

A chain is made by connecting n links. The lengths of different links are independent and uniform over [50,70] millimeters. What is the expected value and variance of the length of a chain obtained by ...
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1answer
669 views

Expected value of sum squared is sum of expected value squared?

Consider the following expression: $ X \sim BIN(1, p) $ $ Var(\bar{X})=Var(\frac{\sum_i{X_i}}{n}) = \frac{1}{n^2} Var(\sum_i X_i) = \frac{1}{n^2} \left( \sum_i E(X_i^2) - ( \sum_i E(X_i) )^2 \right) ...
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1answer
248 views

SVD for Non-Square matrices?

Is QR decomposition a pre-requisite for SVD decompostion of non-square matrices? I have been unable to find a clear cut answer to this question. I will be grateful for a response.
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1answer
176 views

how to prove binomial through bernoulli indicators ??

how to prove binomial through Bernoulli indicators? is it x-bernoulli (P) y=x1,x2,...,xn. where xi is the independent variable bernoulli gives y-Bin(n,p)?
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1answer
286 views

Minimize the sum total of Type I and Type II errors

X is a random variable with density $f(x)=\theta e^{-\theta x}, x>0$. (So X is an Exponential($\frac{1}{\theta}$) distribution.) Consider $H_0: \theta=1$ versus $H_1: \theta=\frac{1}{2}$. I need ...
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1answer
22 views

Bound for normal distribution

suppose $X$ is a standard normal distribution then what is the bound for $Pr \{|X|\leq \epsilon \} $, where $\epsilon \geq 0.$
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1answer
128 views

Inverse transform sampling

I know the basic idea is to generate a random number from $U(0,1)$, find the inverse cumulative distribution function $F^{-1}$ and then take $x = F^{-1}(U)$. If you were plot a histogram of say 1000 ...
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1answer
141 views

Odds of winning a race given the odds of each runner beating each opponent?

Say we have three runners: A, B and C, and we have the probability of each runner beating each individual opponent: A before B: $0.68$ A before C: $0.42$ B before A: $0.32$ B before C: $0.30$ C ...
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1answer
72 views

How to calculate the following variance?

I want to calculate the expectation and variance in the following scenario: $w$ is my initial wealth With probability $0< q_i <1$ with$ i \in\{a,b,c\}$ I lose $a,b$ or $c$ repectively. $U()$ ...
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1answer
74 views

Simple Linear Regression

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|} \text{Obs}\# & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\\hline X & 5 & 8 & 10 & 4 & 5 & 12 & ...
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1answer
132 views

“Practice Exam” probabilities

A practice exam with 6 questions is prepared. 3 of which will be very similar to ones on the actual exam. a) How many possible ways can the exam be written? (i.e. how big is the sample space of ...
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2answers
84 views

Bernoulli Random Variable

Let $X$ be a Bernoulli random variable with probability of success $p$. Answer the following questions. (i) Derive the formulas for the mean, the variance, and the standard deviation of X. (ii) ...
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1answer
324 views

Let $X$ be the total number of heads and $Y$ be the difference between the total number of heads and the total number of tails. Find $\rho(X, Y )$

A coin (with probability of getting head equal to $p$) is tossed twice. Let $X$ be the total number of heads and $Y$ be the difference between the total number of heads and the total number of tails. ...
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2answers
26 views

Probability question of common 1s

Two random 3000 bit arrays each with exactly two ones(1), what is the probability that they have atleast one bit in common?
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1answer
362 views

Poisson distribution using Chebyshey's inequality

Let $X$ have a Poisson distribution with $\mu = 100$. Use Chebyshev's inequality to determine a lower bound for $\Bbb{P}(75 < X < 125)$.
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538 views

to find the probability of a biased coin?

my ques is a bit different one, i don't want to know the probability of throwing coin n times or knowing the probability of biased flipping x times. i want to know the probability p of head, if a coin ...
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1answer
119 views

Does changing two events to mutually exclusive, change their probability?

Does changing two events, $A$ and $B$ for example, to mutually exclusive in a new/altered situation, change their probability from the precondition? Would the $P(A)$, probability of $A$, now equal the ...
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1answer
70 views

limit distribution $Y_n=\frac{1}{n}\sum\limits_{i=1}^n (X_i-10)^2$

Suppose $X_1,X_2,\ldots,X_{n}$ be a random sample of $B(20,.5)$. How can find limit distribution $$Y_n=\displaystyle\frac{1}{n}\sum_{i=1}^n (X_i-10)^2$$
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379 views

how many elements are in A? (sets)

Five applicants (Jim, Don, Mary, Sue, and Nancy) are available for two identical jobs. A supervisor selects two applicants to fill these jobs. Let A denote the set of selections containing at least ...
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1answer
308 views

which of the following random variable is continuous

on a quiz made ​​to any contestant 5 consecutive questions. Has in his possession 20 seconds to answer before proceeding to the next question. if he answer correctly before the 20 seconds recorded ...
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2answers
108 views

Proving that the estimate of a mean is a least squares estimator?

I think this is a really simple question so please bear with me -- I just had my first class in regression and I'm a little confused about nomenclature/labeling. Does anyone recommend some good ...
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1answer
478 views

How do calculate the p-value bounds on Chi Square?

I'm presented with the question: Suppose that we are testing H_0: σ^2 = σ_0^2 versus H_1: σ^2 > σ_0^2 with a sample size of n = 15. Calculate bounds on the P-value for the following observed ...
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1answer
100 views

Checking whether a maximum likelihood estimator is biased

So I have a Poisson distribution: $ V \sim \operatorname{Po} \left({\rho v}\right) $ and I've calculated the maximum likelihood estimator $ \widehat{\rho} = \dfrac{\overline{v}}{v} $ from ...
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4answers
2k views

Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial

The problem: X is a binomial random variable, find $E[\frac{1}{X+1}]$ n and p are not given PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$ Expected value is $\sum{x_ip(x_i)}$ But ...
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1answer
479 views

What is this sentence translated to english: God does not care about our mathematical difficulties. He integrates empirically [closed]

What is this sentence translated to english: God does not care about our mathematical difficulties. He integrates empirically.
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3answers
97 views

Are there reasons to consider complex valued random variables? [closed]

In all examples I know, we always work with real valued random variables. Why don't we ever consider complex valued random variables? Can anyone give an example of such a thing? It seems that if ...
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1answer
316 views

Probability of secretary making 4 or more errors on a page [closed]

I have this problem, and I want to figure out how to do it, or at least figure out the subject that it deals with. A secretary who only does word processing makes $2$ errors per page when typing. ...
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votes
2answers
58 views

Operations on distributions

Say we have two r.v X and Y which are independent and differently distributed ( for e.g X follows a bell curve and Y follows an exponential distribution with parameter $\lambda > 0$ What are the ...
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1answer
976 views

Proof that mutual statistical independence implies pairwise independence

This question about pairwise vs. mutual relations is related some extant questions: here and here. Kobayashi, Mark & Turin's Probability, Random Processes and Statistical Analysis, 2012, states ...
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1answer
641 views

Statistics Help

I have some questions that I'm not sure of. Any help is appreciated. 48.7% of Americans have brown eyes. A convention has 5000 people in attendance. Find the mean (expected value),the standard ...
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2answers
138 views

Please solve for X

I messed up the equation last time I asked this - Can someone please solve this function for X? $Y = \displaystyle 0.5\:a\:\text{erf}\left(\frac{x-b}{c\sqrt{2}}+.5\right)+d$ When I solve for Y with ...
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1answer
67 views

Excel Problem-Correlation [closed]

A rich stock picker wants to decide between two stocks. One stock will pay a dividend of 100,000 its first year. After that, the dividend options look like this: a1=1, 1ST YEAR DIVIDEND + 30,000 ...
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votes
2answers
81 views

Why is one of the terms ignored when finding the variance of a sample of 5 sales?

$2,4,5,2,7$ Mean: $4$ Variance: $\large\frac{4+0+1+4+9}{5}=\frac{18}{5}$ Standard deviation: $\large\sqrt{\frac{18}{5}}=1.90$ The answer ignores the $(4-4)^2$, so that is ...
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1answer
44 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 ...
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1answer
888 views

Maximum Likelihood Estimator of $f(x;\theta) = (1/\theta)x^{(1-\theta)/\theta} $

Let $f(x;\theta) = (1/\theta)x^{(1-\theta)/\theta} $$\hspace{20 mm}$ $ 0 <x <1 ,\hspace{5 mm} 0 <\theta<\infty$ I need to show that the maximum likelihood estimator of $\theta $ is ...
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1answer
79 views

Economics and Statistics

Let $Y$ be a random variable and write $\mu = E[Y ]$. Show that $E[Y −\mu]=0$. and Let $X$ and $Y$ be random variables. Prove that $\text{Cov}[X, Y ] = E[X(Y − E[Y ])]$.
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1answer
284 views

Sufficient statistic and MLE problem

Let $X_1, \ldots, X_n$ be i.i.d. random variables with pdf $\Theta x^{-2}$, $0 < \Theta \leq x < \infty$. Find a sufficient statistic for $\Theta$. Find the MLE of $\Theta$ Any ...
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1answer
2k views

Statistics, calculating mean and standard deviation

IF the mean of a set of scores is 80 and standard deviation is 10, explain the effect on mean and standard deviation in case following changes are made to each of the scores 1) adding 5 2) ...
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votes
1answer
75 views

Markov/Chebyshev's inequality Problems

Let $X$ and $Y$ be two random variables for which $ E(X)=75 $, $ E(Y)=75 $, $\mathrm{var}(X)=10$, $\mathrm{var}(Y)=12$, $\mathrm{cov}(X,Y)=-3$ (i) Find and upper bound to $P(|X-Y| \ge ...
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1answer
81 views

Two Top Economist getting 9/10 correct

Suppose that we now have twenty economists instead of just one, each of whom makes their predictions based on the toss of a fair coin. what is the probability that the second most successful of the ...
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2answers
127 views

Expression of Discrete and Continuous random variables.

Let $X$ be a discrete random variable. And let $Y = cX$ for some constant $c$. How can you express the distribution of $Y$ in terms of the distribution of $X$? Let $X$ be a continuous random variable ...