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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What is the meaning of $E[g(x)]=Pr(|x| >\varepsilon)E[g(x)|g(x) > g(\varepsilon)]+Pr(g(x) \leq\varepsilon)E[g(x)|g(x)\leq g(\varepsilon)]$

Any body knows the meaning of this expectation ($E[g(x)]$) form? $E[g(x)]=Pr(g(x) >\varepsilon)E[g(x)|g(x) > g(\varepsilon)]+Pr(g(x) \leq\varepsilon)E[g(x)|g(x)\leq g(\varepsilon)]$
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5k views

Calculating variance of Discrete Uniform distribution when its interval changes

I am not excited about grading exams. I would rather jam a dull stick into my leg. I will therefore randomly assign your grade by picking an integer uniformly from 77 to 100 (inclusively). ...
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36 views

Finding collinearity among variables

I have been reading about how (multi)-collinearity among predictor variables can be determined by looking at the condition number, or smallest eigenvalue, of the covariance matrix. My question is, if ...
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261 views

Chebyshev's inequality question

Question: Let $X$ be a random variable. mean= 80 , std deviation ($\sigma$) = ? (unknown) find std deviation for which $P(75 < X < 85) \geq 0.9 $ My approach is .. $ P( |X - 80| \lt ...
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119 views

Poisson random variable with parameter $\lambda>0$

I am currently studying Poisson distribution. I could understand \begin{align} \mathrm{E}(X) &= \sum_{k=0}^{\infty} \frac{k \lambda^k e^{-\lambda}}{k!}= \\ &= e^{-\lambda} ...
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135 views

square root estimator

Let's say we want to do an estimation using iid samples $X_i, i=1,2,3,..., N$ the following formula, $$\hat{X}_1 = \frac{1}{N}(\sum_i\sqrt{X_i})^2$$ square sum of square roots. This form also seems ...
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123 views

Expected Value in Lottery Game

In a lottery game, a person selects a three digit number. If the person plays for \$1, he can win \$500 . Find the expectation. In the same lottery, if a person boxes a number , he will win \$80. ...
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How to resolve this probability question?

today I was answering a exam and I get a problem which I have no idea how to resolve it. Here is the announcement $500$ ​​people attend a nightclub. Those who are members of the club pay 14 ...
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63 views

Does $E(X_1) \leq E(X_2)$ implies $E(f(X_1)) \leq E(f(X_2))$ for a monotone increasing function $f\colon R \mapsto R^{+}$?

For two random variables $X_1$ and $X_2$, does $\mathrm{E}(X_1) \leq \mathrm{E}(X_2)$ implies $\mathrm{E}(f(X_1)) \leq \mathrm{E}(f(X_2))$ for a monotone increasing function $f\colon \mathbb{R} ...
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35 views

Why is the sample correlation coefficient not $1?$

A reasonable value for the sample correlation coeffcient $\rho$ between daily maximum tem- peratures and daily ice cream sales would be $A) 0$ $B) 1$ $C) 0.7$ $D) -0.7$ ...
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214 views

Probability of $3$ people in a room of $5$ having the same birthday

What is the probability that, amongst five people in a room, three have the same birthday? I was wondering about this twist on the birthday problem. I am not a major stats guy so I want your help.
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761 views

Why do we use a $z$-test rather than a $t$-test when estimating an appropriate sample size?

I'm kinda puzzled on one point. In our stat class, we are taught to use the Student $t$ distribution to find confidence intervals for normally distributed data, as blindly using the normal ...
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91 views

Need help with derivation of conditional expectation

The following is taken from the book "Mathematical Statistics for Economics and Business": \begin{align*} E\left.\left( \left[ Y-h(x) \right]^2\ \right\vert\ x\right) =& E\left.\left( ...
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396 views

Fitting a polynomial + exponential curve of a given form to data

I have got a number of data sets of some parameter $m_x$ against an independent variable $x$. Through each of the data sets I need to best fit a curve of the form $A + Bc^x$ such that $A$, $B$ and $c$ ...
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172 views

Confidence Interval Questions

I'm having trouble with this question and was wondering if anyone could help me out. It goes like this: The manufacturer of a certain type of product claims that the machine that fills packages of ...
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28 views

A small calculation .

How $\sum_{k=0}^n (-1)^n\times(-1)^{n-k}=\sum_{k=0}^n(-1)^k$ i got it $\sum_{k=0}^n(-1)^n\times(-1)^{n-k}=\sum_{k=0}^n(-1)^{2n-k}$ And is that $\mathbb E[\mathbb E(X)]=\mathbb E(X)$ ?
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125 views

How to calculate $\Pr[\max(X,Y)<4]$?

Suppose the joint PDF of X,Y is $f(x,y)=1/40$ and $0 < x < 5$ and $0 < y < 8$. How to calculate $\Pr[\max(X,Y)<4]$?
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181 views

What is some application of variance in actuarial science/insurance risk?

What is some application of variance in actuarial science/insurance risk? I learn that a lot of applied math book of actuarial science have variance of probability distribution frequently. Don't ...
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36 views

Is that Probability function only for discrete case?

Most of the books and sites define Probability function for discrete case that is they use the term as the synonym of Probability mass function. Is that Probability function define for only ...
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50 views

what does this really mean

what exactly does this expression mean, i keep seeing it in statistics but i never really understood what its supposed to be, is it another way of writing the variance ...
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58 views

MLE estimation for number of customers.

A clerk in a shop has noticed two customers arrived at the shop between 12:00 and 12:45. Another clerk noticed only one customer between 12:15 and 13:00. Assuming a Poisson distribution on the number ...
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86 views

Cumulative distribution function of the generalized beta distribution.

Suppose $Z$ has a beta distribution on the interval $(0,1)$ and its probability density function is $f_Z(x)$. I know that the cumulative density function is, $$F_Z(x) = \mathbb{P}(Z \leq x) = ...
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265 views

Naive bayes: Log odds derivation

How does one go from line 17 to 18 in the below picture? i.e. conversion to linear function of the input variable. Source: http://pages.cs.wisc.edu/~jerryzhu/cs769/nb.pdf
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Statistic - different question (person's breathing )

Research psychologists are interested in finding out whether a person's breathing patterns are affected by a particular experimental treatment. To determine the general respiratory patterns of the ...
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36 views

How to prove statistical hypothesis?

I developed a caching method. I took 100 experiments and got that hit ratio is not less than 75%. Now, I want to prove that my method with some probability gives hit ratio not less than 75%. How ...
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447 views

Uniform distribution joint $\to$ marginal

Let vector $(X,Y)$ have a uniform distribution on the set $N = \{ (x,y): x<1,y<1,1<x+y\}$. Determine distribution $X-Y$. So far I've thought of this: \begin{align} P[X | Y=y] &\sim ...
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154 views

What is the purpose to define central moment?

What is the purpose to define central moment? I searched the google and all i could find is bunch of properties
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1k views

Covariance of order statistics (uniform case)

Let $X_1, ..., X_n$ be uniformly distributed on $[0,1]$ and $X_{(1)}, ..., X_{(n)}$ the corresponding order statistic. I want to calculate $Cov(X_{(j)}, X_{(k)})$ for $j, k \in \{1, ..., n\}$. The ...
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425 views

question on five number summary & quantile.

i know that in five number summary : 25% of a data set lies between Min & 1st quartile. 50% of a data set lies between Min & 2nd quartile, that is, Median. 75% of a data set lies between ...
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530 views

please prove the following proof related to F distribution.

Suppose $S_1^2$ and $S_2^2$ are two independent unbiased estimate of the common population variance $\sigma^2$ from two random sample of sizes $n_1$ and $n_2$ respectively. Then show that ...
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Expectation Values

Suppose that $\{X_n\}_{n\ge1}$ take values $-\dfrac{1}{2}$ and $\dfrac{1}{2}$ with probability $a$ and $1-a$ respectively and $0$ otherwise. Suppose further that they are independent and discrete and ...
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38 views

Calculating fairness in a competition across offices

We want to run a fitness competition in our company using pedometers to determine which office location is the healthiest. The problem is, we can't use total number of steps, because each office has a ...
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37 views

Manipulating an equation in statistics

We have $y = X\alpha + \xi$, where $\xi\sim N(0, 1)$ and $\xi = (\xi_1\,,\xi_2\,,\xi_3\,,\xi_4)^T$. Moreover $X$ is a 4x4 design matrix such that $x_i$ is a column 4x1 column vector made up of 1's, ...
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832 views

Find the maximum likelihood estimator for $\theta$ when $f(x)=2\theta^{-2}x, 0\leq x \leq \theta$

Find the maximum likelihood estimator for $\theta$ when $f(x)=2\theta^{-2}x, 0\leq x \leq \theta$. This should be a really easy question but I somehow cannot seem to get the right answer. My ...
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843 views

average is higher than percentile 90 ?…

I have a few hundreds of results from a test im running, and with a utility I have that analyzes these results I can see that the Average result value is higher than the percentile 90th value... This ...
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81 views

Convergence of probability densities

I appreciate if you can give me some guidance on how to approach this question: Suppose $f_n(x) \text{ and } g(x)$ are densities such that for all x, $f_n(x) \rightarrow g(x)$ as $n \rightarrow ...
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95% confidence interval for the mean becomes irrelevant when t-test is insignificant?

For the results I have been given, I have been asked to interpret the 95% confidence interval for the mean. However, the t-test shows a value more than .05, meaning that the differences I am exploring ...
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697 views

Rat in Maze Probability

I am trying understand what am I missing in my way of solving rat in maze problem... The question and solution is given in this link http://www.ams.sunysb.edu/~jsbm/courses/311/rat-in-maze.pdf ...
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72 views

Find sample variance from sample mean and mean of squares

I have ten measurements like this: and I have calculated the sample mean E(X) = 10.03, and the average of squares E(X^2) = 100.679: Now I want to calculate the sample variance. I use the ...
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4k views

Probability of hitting the bullseye in darts

I throw darts repeatedly. Assume that on each throw I have a 1% chance of hitting the bullseye, independently of all other throws. (Note that this implies for example that repetition doesn’t help my ...
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loaded die and probability

A loaded die has probabilities 1/21, 2/21,3/21, 4/21, 5/21, 6/21 of showing 1,2,3,4,5,6. a) What is the prob of throwing 2 3's in succession? So, is the answer (3/21)^2? b) What is the prob of ...
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68 views

Finite expectation

Let $X_1,X_2,...$ be i.i.d with mean $\mu$. Let $T$ be a stopping time with respect to $X_,X_2,...$ with $E(T)<\infty$. Show that $E\left(\sum_{n=1}^\infty |X_n|I\{T\ge n\}\right)<\infty$. My ...
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497 views

Probability of missiles on target

The captain of a Navy gunboat orders a volley of 25 missiles to be fired at random along a five-hundred-foot stretch of shoreline that he hopes to establish as a beachhead. Dug into the beach is a ...
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Two Tail Hypothesis test

Use a .01 significance level, a two tail test and the following data to test the hypothesis the proportion of seniors who support Obama equals the proportion of freshmen who support Obama. ...
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74 views

Strategy to identify the best coin with highest probability to produce head

With n coins, each has a certain probability that generates a head. Using as small number of tosses as possible, how to find the coin that has the highest such ...
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181 views

Infinity/exponential problems.

I want to evaluate $$\int_{0}^{\infty} e^{(i\alpha-1)x}\,\mathrm dx,$$ where $i$ is the imaginary number. $$\left [ \frac{e^{(i\alpha-1)x}}{i\alpha-1}\right]_0^{\infty}$$ At this point, I beleive, ...
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49 views

What is the distribution of $\bar x$?

Let $x_1,x_2,\ldots,x_n$ be a random sample from a normal distribution with mean $\mu$ and and variance $\sigma^2$. show that $E[\bar x]=\mu$ and $V[\bar x]=\sigma^2 /n$, where $\bar x = ...
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532 views

Find the conditional probability density function.

If I am given $X$ that follows an exponential distribution with mean $m$ and $Y$ that follows a poisson distribution with mean $n$, how can I use them to find the conditional probability density ...
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956 views

How to work out percentage/probability of recieving these marks

I have a couple of maths question I want to ask, I am a computing student and I really want help with the answers to the following: I am trying to state that if a test has more questions, then the ...
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888 views

Statistics question on probability

Suppose that one hundred fair dice are tossed. Estimate the probability that the sum of the faces showing exceeds 370. Include a continuity correction in your analysis.