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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1answer
8 views

Correlation Coefficient of data on $y = mx + b$

Suppose n cases of data on $x$ and $y$ fall exactly on the line $y = mx + b$. Compute the value of $r$. (Hint: In any of the formulas for $r$, eliminate all $y$ in favor of $x$.)
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1answer
16 views

Chance of Winning Sale

If there is a 25% chance of winning a sale but I suck at selling so my chances of making this sale is 20%, what is my probability of winning this particular sale?
0
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28 views

EV of sequences of coin flips

I have a biased coin $C$ where $P(H) = p$ and $P(T) = 1-p$. This coin could be tossed an infinite number of times and the results will be recorded. I want to calculate the probability of a given ...
0
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2answers
19 views

when tossing a coin ten times, what is the probability of an outcome which has a string of 3 or more heads as well as a string of 3 or more tails?

here is an experiment from my Stat textbook "Try this experiment: Write down a sequence of heads and tails that you think imitates 10 tosses of a balanced coin. How long was the longest string ...
1
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2answers
28 views

Frequency of the most frequent word using Zipf's law

I modeled my corpus using Zipf's law, so that: $$f_r=c/r$$ Where $f_r$ is the frequency of a word at rank $r$. I calculated $c=90000$. How can I calculate the frequency and the occurrence for the ...
2
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0answers
53 views

Help to understand the Monty Hall Problem [on hold]

I have read the Monty Hall Problem and explanations on Wikipedia several times and fail to grasp how exactly 'conditional probability' is being realistically applied. More specifically, my logical ...
0
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0answers
12 views

Generalized least squares equation project [on hold]

I am trying to come up with a project incorporating the generalized least squares equation. Does anyone have any ideas of what I could do or where I may be able to find data for this particular ...
1
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0answers
13 views

Help simplifying equation (sample variance is an unbiased estimator)

I would like to know how you get from this to $\sigma^2$. I tried to simply as much as I could and got
3
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1answer
31 views

UMVUE of $ \frac{1}{\theta} $ coming from $ f(x) = \theta x^{\theta - 1} $.

Let $ X_1,\ldots,X_n $ be i.i.d. sampled from the distribution $$ f(x;\theta) = \theta x^{\theta - 1}, $$ where $ x \in (0,1) $ and $ \theta > 0 $. Show that $$ T(x_{1},\ldots,x_{n}) = - ...
-1
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0answers
26 views

Probability and stats question [on hold]

Im kinda confused on how to approach this question and how to solve for it. i know there are multiple ways but i was wondering if anyone could help me. thanks 1.) It appears that the mean commuting ...
0
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0answers
11 views

Understanding two sided t-test

Assume we have two search engines, A and B. I get a list of scores for 10 different queries. Now, I model this with a t-test in order to test significance. These are my hypothesis: $H_0: B-A=0$ ...
0
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0answers
8 views

How to estimate theta by method of moment estimator and determine maximum likelihood estimators

I have not learnt the POINT ESTIMATION but I have had time to go through some written notes on how to estimate theta1 and theta2. I would love to do this question but I am lacking some knowledge on ...
0
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2answers
30 views

Finding average using central limit theorem.

I'm have a lot of trouble trying to wrap my head around how exactly to use the central limit theorem in this question. Image a gambling casino consisting of 100 busy roulette tables. Suppose that ...
1
vote
1answer
23 views

Fifty men and thirty woman are lined up at random. How do I find the expected number of men who have a woman standing next to them.

Fi.fty men and thirty woman are lined up at random. How do I find the expected number of men who have a woman standing next to them.
2
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1answer
18 views

How would one estimate the number of commercial airplanes bought in the US every year? [on hold]

I am not sure where to start with this question. Do you start by estimating number of major airports? How many airplanes decommissioned/and new ones bought?
1
vote
1answer
23 views

What is the expected value of the highest of N independent draws from the unit uniform distribution?

I have a seemingly innocuous problem that I can't seem to wrap my head around. The following is mentioned in passing on one of my lecture slides, but when I try to arrive at the same conclusion I get ...
0
votes
2answers
24 views

How do the statistical calculation according to X people? [on hold]

It's common to read something like "for 10 people, 5 prefer something" (5 people this total of 10 people).Eg: 8 in 10 people over 50 years old use the Internet ...
0
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0answers
35 views

Help: Studying A-Level Mathematics [on hold]

Although I am a latecomer at the age of 21 years of age, I have enrolled in self taught mathematics A-level with "Edexcel" both mathematics & further mathematics. I am in need of help with ...
0
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0answers
20 views

Product of (multivariate) Gaussian densities

One can frequently read that the product of two multivariate Gaussian pdf f1(x) and f2(x) is itself a Gaussian, with parameters as defined for example in: ...
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0answers
10 views

Xi~Poi(theta) and let Tau(theta)=e^theta. a) find the MLE of Tau(theta). [on hold]

consider a random sample from a poisson distribution, Xi~Poi(theta) and let Tau(theta)=e^theta. a) find the MLE of Tau(theta). b) Is this MLE an unbiased estimator of Tau(theta) c) Find the CRLB ...
0
votes
1answer
9 views

Normal Distribution: Statistics

I'm having a lot of trouble trying to remember the formulas on how to calculate these questions. Any help would be great. An automobile insurer has found that repair claims are Normally distributed ...
0
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1answer
16 views

Prove that a symmetric distribution has zero skewness

Prove that a symmetric distribution has zero skewness. Okay so the question states : First prove that a distribution symmetric about a point a, has mean a. I found an answer on how to prove this ...
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0answers
16 views

Random Variables and Statistic

I'm studying Statistical Inference by Casella and I'm confused with the definitions of random variable & statistic. So let we have the probability space $(\Omega, F, P)$ where $\Omega$ is the ...
0
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2answers
20 views

Expectation of a discrete variable [on hold]

Suppose $P(X=n)=kq^{n-1}$ where k is a constant. Then how do I find a value for the expectation of X?
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0answers
15 views

Find upper bound of probability value using Chebyshev's inequality [on hold]

Given density function of random variabel X is f(x) = 1/(2√x), for -√3 < x < √3. Use Chebyshev's inequality to find upper bound of probability value P(IxI≥3/2).
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0answers
10 views

Find lower bound of probability value using Chebyshev's inequality

Given density function of random variabel $X$ is $f(x) = 3x^2$, for $0 \lt x \lt 1$. Use Chebyshev's inequality to find lower bound of probability value : $P(5/8 \lt x \lt 7/8)$ $P(1/2 \lt x \lt 1)$ ...
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votes
1answer
11 views

Find the probability P(x is even) of given cumulative distributive function [on hold]

Given cumulative distributive function (CDF) $F(x) = 1 - (1/2)^{(x+1)}$ for $x = 0, 1, 2, ...$ Find the probability value P(x is even).
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0answers
19 views

expectation calculation problem small problem

a Continuous, positive random variable X, whose PDF is proportional to $(1+x)^{-4}$, where $0<x<\infty$, determine $E(X)$ i tried to solve it directly by integrating from 0 to infinity to get ...
0
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0answers
13 views

Bayesian update multivariate normal based on one-dimensional signal: simple rule

Is there a simple rule to update the linear combination of normal distributions based on a one-dimensional signal? The unconditional joint density of $(\eta,\theta)$ is multivariate normal ...
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2answers
23 views

expectation calculation problem

I got the answers for this and i know its 1.05 but the way it explains is very difficult to understand so im seeking for some help here. A system made up of 7 components with independent, identically ...
0
votes
3answers
24 views

Applying the basic formula for binomial distribution

I'm pretty confused on how this works. In my class my teacher states that: Let $X$ be a random variable with $S_X = \{0,1\}$. $X$ follows a Bernoulli distribution if $P(X = x) = p^x(1-p)^{1-x}$ for ...
1
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1answer
20 views

Distribution of random variables (normal and standard normal)

Suppose that $X_i \sim N(\mu, \sigma^2)$ for $i = 1, \ldots, n$ and that $Z_i \sim N(0,1)$ where all of the random variables are independent. Denote $s^2_Z$ as the sample variance of $Z_1 , \ldots, ...
0
votes
1answer
58 views

How to find $E[Y|X=1]$?

A fair die is repeatedly rolled. Let $X$ and $Y$ denote, respectively, the number of rolls required to obtain a $1$ and a $2$. How do I find $E[Y|X=1]$? edit: for using this I got 1*6 = 6 total ...
0
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0answers
14 views

Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
0
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0answers
4 views

A mix between the Horvitz-Thompson and ordinary estimator

I have asked this question on mathoverflow, but got no answer. Here I have corrected some mistakes and wish to hear any ideas that may bring at least numerical result: The data I have two samples: ...
0
votes
1answer
16 views

Selection of Distribution model

An expressed parcel delivery company offers a First Class service for which it is promised that 80% of all parcels are delivered within 24 hours of dispatch. It is suspected that the true successful ...
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0answers
41 views

Standard deviation and mean question [on hold]

The pulse rates of healthy adults are normally distributed with mean 72 and standard deviation of 5. i) Approximately ninety-five percent of the adult population can be said to have their ...
0
votes
1answer
32 views

formal proof that p-values are uniformly distributed

I'm trying to prove that $p$-values under the null hypothesis are uniformly distributed in $[0, 1]$ for an absolutely continuous test statistic $X$. Proof: By continuity of $F_X$, it is sufficient to ...
0
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0answers
21 views

How to get more profit in stochastic process?

Suppose there is a system, for each step, I cost something but I didn't know how much I cost, and the system return to me something, which follow Guassian distribution and the expectation is what I ...
0
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0answers
23 views

Deriving a formula for a confidence interval

Derive a formula for a $(1-\alpha)100\%$ C.I. for $\mu_x -\mu_y $ for data that has the following properties: A random sample $X_1,X_2...X_n \ are \ i.i.d ~N(\mu_x, \sigma^2 ) $ Another random ] ...
0
votes
3answers
20 views

How to prove expected value of uniform random variable?

I tried this: $$\int_a^b t~dt = \frac{t^2}{2}\Big]_a^b = \frac{b^2-a^2}{2} = \frac{(b+a)(b-a)}{2}$$ Isn't it supposed to be $\frac{b+a}{2}$ or something like that? Obviously if I multiply the ...
0
votes
1answer
22 views

computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
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0answers
86 views

Is the Monte Hall Problem a Ludic Fallacy? [on hold]

---- EDIT #2 ---- I decided to "undelete" my post and to undo my "Edit #1" so people can see my original post. ---- My Very LONG Original Post ---- Suppose Fat Tony is on a game show, and he's given ...
0
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2answers
27 views

Confusion with Z-Score

Having some issue with the concept of Z score. When exactly do I use $Z = \frac{\bar X - u}{\sigma}$, and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$. I get very confused ...
0
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1answer
29 views

4 cards are shuffled and placed face down. Hidden faces display 4 elements: earth, wind, fire, water. You turn over cards until win or lose.

Question: 4 cards are shuffled and placed face down in front of you. Their hidden faces display 4 elements: water, earth, wind, fire. You turn over cards until win or lose. You win if you turn over ...
0
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1answer
63 views

incorrect rejection of a true null hypothesis? [on hold]

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
0
votes
1answer
26 views

Method of moments estimation for $\theta$

I read one example in my notes, but I couldn't find out how the answer in my notes is derived. If $x_1,...,x_n$ are realizations of a random variable distributed with the following PDF: $f(z; ...
2
votes
0answers
27 views

How sample size affects confidence interval.

Suppose the weight of n primary one students has sample mean of 20KG. If n = 40, a certain percentage of confidence interval for the population mean is (15.5,24.5). Find the confidence interval if we ...
0
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0answers
16 views

F-test and T-test produce the same results

I am modelling a stochastic process by two different methodologies and I expect the results of each to be normally distributed with identical means and stdevs. To test that the distributions after ...
0
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1answer
50 views

Finding the probability using a normal distrubtion.

I have a stats question that says, "An airline flies airplanes that hold 100 passengers. Typically, some 10% of the passengers with reservations do not show up for the flight. The ...