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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From correlation coefficient to conditional probability

In the best-selling book Thinking Fast and Slow (p. 205), Daniel Kahneman (a Nobel Prize winner in Economics) makes the following claim: 'Suppose you consider many pairs of firms. The two firms in ...
3
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1answer
15 views

co-variance between a sample from normal distribution and the sample mean?

$X_1$ is a sample from a normal distribution with mean$=\mu$ and variance $= 1$. The joint distribution of $X_1$ and the sample mean is bivariate normal. I need to find the conditional distribution of ...
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1answer
29 views

Clarifying the assumptions about a paired t-test

I've wrote my question in red ink (see links). There are two questions that I have. Primarily I want to know why they concluded that "there is some evidence that there is some difference in mean ...
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2answers
37 views

Choosing C so that c1Y1+c2Y2 is an Unbiased Estimator of Θ

Let Y1 and Y2 be two unbiased estimators of Θ let Var(Y1)= 4Var(Y2) and the correlation coefficient between Y1 and Y2 be -.5 find constants c1 and c2 such that c1Y1 + c2Y2 is an unbiased estimator of ...
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0answers
5 views

How can I decide, using a strictness parameter, whether a collection of values are “equal enough” to be trusted?

Suppose I have a set of N scalar data points which I don't entirely trust the measurement of but can't repeat the measurement myself, nor is there any way to generate further data. For example, I am ...
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0answers
13 views

polynomial chaos expansion: linear combination properties?

I'm dealing with polynomial chaos expansion, for finite support specifically. Assume $X$ and $Y$ are r.v.'s whose the inverse CDFs expressed as $$ F^{-1}_X(x) = \sum_{j=0}^{N} s_j^{(X)} \psi(\xi)$$ ...
2
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1answer
984 views

How do I compute “AUC” Area under the curve number, if all I have are my TPR and FPR values?

I am trying to rank my neural network, which is trained for binary classification. That is, given a set of input signals, it outputs either a 1 or a 0. I have a training set, where I have the actual ...
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2answers
28 views

Binomial Distribution with Coupon Collecting

Suppose I have 6 coupons found in a box of cereal that are randomly distributed throughout the boxes. If I wanted to find the probability that it takes at most 8 boxes of cereal to find all the ...
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0answers
17 views

What does this question mean? The Wald test

Looking at a past paper without soltuions, I am unclear of what is being asked. Context $x_1...x_n$ denotes a random sample from a normal distribution $N(\mu,\theta)$. After I've obtained the ...
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0answers
21 views

How Much of a Distribution has One Seen After N Samples with Replacement?

I frequently run into this question while modeling processes. I am wondering if there is a general solution or approximation. The question I run into is: For a given distribution with a finite ...
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0answers
17 views

statistical likelihood of a sudden jump in falure rate

This question relates to a problem I have with a circuit board manufacturing line. We have seen a sudden jump in failure rate of a particular component part way through a production run of 211 circuit ...
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0answers
35 views

Why is the following expectation inequality true?

If $X_1$ and $X_2$ are random variables, why is the following inequality true: $$|\mathbb{E}X_1 - \mathbb{E}X_2| \leq \mathbb{E}|X_1 - X_2|$$
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0answers
16 views

logarithm of odds score at test value of 0

In my human genetics class, we learned to compute a logarithm of the odds (LOD) score as follows: $LOD=\log_{10}{\frac{\theta^r(1-\theta)^{n-r}}{0.5^n} }$, where $\theta$ is a test value for ...
0
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0answers
10 views

Bayes Rule and Multivariate Normal Estimation

This is an exercise in this pdf file http://statweb.stanford.edu/~ckirby/brad/LSI/chapter1.pdf and how can I show that by using Bayes Rule?
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0answers
9 views

Nonlinear regression output in R

Suppose one is interested in doing a nonlinear curve fitting procedure such as $Y=AX^B$ where $a,b \in \mathbb{R}$. If the regression were linear, one usually observes the standard error of the ...
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1answer
34 views

Spearmans Rank, why does it work?

Looking at spearmans rank, can someone explain how the forumula works, is their anything intuative about it?
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1answer
23 views

Can a prediction interval be interpreted as a probability?

Suppose I find a 90% prediction interval for some data distribution. This implies that if I sample large enough data from this distribution, then 90% of such data will lie inside the prediction ...
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0answers
10 views

Accessing the Validity of a Nonlinear Curve Fitting

Suppose I do a nonlinear regression with the following form $Y = a*X^b$ where $a,b \in \mathbb{R}$. I do this is the R programming language with: ...
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1answer
31 views

Basic questions concerning sample means and distributions

I have the following questions: Is The value of the sample mean always the population mean $\mu$, in any sample? I am confused about whether or not it is. Is the sampling distribution of the sample ...
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0answers
8 views

Distribution of sample minimum after bivariate selection (double truncation)

Let $X$ and $Y$ be two RVs with joint distribution $$ (X,Y)\sim \text{Normal}(\mu,\Sigma) $$ Suppose that there is selection on $X$ and $Y$, such that we observe a vector of realisations of $X$, ...
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0answers
16 views

Using CLT to determine the sample size to achieve a given power.

Consider a distribution having a pmf of the form $f(x;\theta)=\theta^x(1-\theta)^{1-x}$ $x=0,1$, zero elsewehre. Let $H_0: \theta=\frac{1}{20}$ and $H_1: \theta>\frac{1}{20}$. Use the Central Limit ...
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0answers
20 views

Unconvinced with the expectcation calculation on this one

I am frustrated with why this is happening. The distribution is truncated Poisson with $\theta$ and $y_1,...,y_n$ observations. If $l(\theta)$ denotes the log likelihood of the distribution, I need ...
0
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1answer
23 views

Does a factorable joint CDF/PDF always imply independence?

Say we have two random variables, x and y, with $F_{xy}(x,y)$ and $f_{xy}(x,y)$ denoting their joint CDF and PDF respectively. If they can be written such that $$F_{xy}(x,y)=G(x)H(y)$$ ...
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0answers
12 views

Monte Carlo with non uniform weighting

So, I just want to check if what is in my mind is in fact true. Assume, that we have a distribution over the whole $\mathbb{Z}^+$, where $p(k) = \gamma_k$. We are interested in approximating $p(v)$ ...
3
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1answer
605 views

Real life math to explore/solve

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
2
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1answer
21 views

Converting Permutations to Combinations: Simple Stats in Practise

In a popular text book there is a question that has bothered me that I am sure is very simple for others and I'm just missing something..... So image $100$ songs and we have $10$ as Beatles songs. We ...
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1answer
15 views

Book recommendations for introductory Bayesian statistics?

Anyone here have some recommendations for a good book introducing the reader to Bayesian statistics? Let me mention my background. My undergraduate majors were in Actuarial Science and Statistics, ...
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2answers
25 views

How do I find the probability of committing a Type II error?

I'm not sure If I understand what (b) is asking. Does it mean that the alternative hypothesis will be p<0.3, p<0.4, and p<0.5? If it is, then I have to find the probability of committing a ...
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3answers
39 views

Uniform distribution and Exponential random variable mix

Here's the question Let X and Y be independent random variables, where X is uniformly distributed over (2,4) and Y is exponentially distributed with mean 3. Find the density of U=X/Y Here's what I ...
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0answers
12 views

Pure jump process

I'm having touble understand the pat of the solution that I have underlined in green for b)
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1answer
28 views

Statistics, two products made of same two materials, max price

I can't solve this problem: You have two products, which consist of two materials. First one has a price of 3 units, second is worth of 2 units. For fist one you need 2 pieces of first material and 4 ...
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0answers
10 views

Does $var(\sum X_i' (X_i\beta + \epsilon_i)\vert X_i \forall i) = var(\sum X_i'\epsilon_i \vert X_i \forall i) $?

As stated in the question, does $$var(\sum X_i' (X_i\beta + \epsilon_i)\vert X_i \forall i) = var[\sum X_i' (X_i\beta) \vert X_i \forall i]+ var(\sum X_i'\epsilon_i \vert X_i \forall i) =var(\sum ...
1
vote
1answer
29 views

Unknown distribution of a random variable

$X_1, X_2, \ldots, X_{400}$ is a random sample from given distribution with median of m ($P(X_i \le m)=0.5$). Calculate $P(X_{220:400} \le m)$. How to calculate that? I am lost with this question. ...
0
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1answer
23 views

What is the average number of random selections it would take to have picked every element of a set and the size of that set, n?

I've been discussing this question with my AP statistics teacher and we're both racking our brains as to how this probability distribution would look. The problem came up when looking at the scenario ...
0
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1answer
16 views

LAD analytical minimization

Is it possible to minimize least absolute deviations analytically? Say given a sample $\{x_i\}_{i=1..n}$ find $$\arg\min_\lambda{\sum_{i=1}^{n}{|x_i-\lambda|}}$$
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0answers
16 views

Maximum Likelihood Estimators when $F(x) = 1 - (b/x)^a$

Suppose that random variables $X_i, \ldots, X_n$ are independent and identically distributed with the CDF: $$F(x|a,b) = 1-\left(\frac{b}{x}\right)^a$$ With the conditions $x \ge b, b \gt 0$ and $a ...
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0answers
9 views

Hat matrix and leverages in classical multiple regression

What is Hat matrix and leverages in classical multiple regression? What are their roles? And Why do use them? Please explain them or give satisfactory book/ article references to understand them. ...
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0answers
14 views

Summation of a fraction containing a summation operator

I came across a proof which had the following sequence: $$\sum_{i=1}^n k_i y_i = \frac{(x_i - \bar x)y_i}{\sum_{i=1}^n (x_i - \bar x)^2}$$ where $$k_i = \frac{(x_i - \bar x)}{\sum_{j=1}^n (x_j - ...
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2answers
43 views

Determine who is the best seller

The numbers below show the number of lollipops Betty and Sharon each month for a total of 12 months or a year. Using the data and plot below, can you determine who is the bestseller? Would it be ...
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0answers
39 views

Ratio of independant standard normal random variables.

I want to solve this question below. But I have no idea how to even start it. Any help would be appreciated.
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0answers
16 views

Convergence in distribution of the following sequence of random variables

$X_n\sim Beta\left(\frac{\alpha}{n},\frac{\beta}{n}\right)$ with $\alpha>0$ and $\beta>0$. Does $X_n$ converge to a distribution?
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1answer
45 views

Please help check the answer for finding the probability of people arriving

People arrive at a variety store once every 10 minutes or so. (a) What is the probability nobody arrives in the next 15 minutes? Ans: 0.223 (b) What is the probability at least 3 people arrive in the ...
0
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2answers
37 views

How to prove that This limit was used in deriving the Poisson density

From my understanding is to use l'hopital's rule is that right? Please help me explain. Thank you!
1
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1answer
202 views

Poisson distribution-Queueing theory

Vehicles arrive at a junction, in order to swing left, create a line queue ( tail) . The number of vehicle follow Poisson distribution. The length of cycle for the traffic light (for left turns ) is 1 ...
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1answer
39 views

Root Mean Square Error - How did he get this number?

So I am studying for a college final exam, and following a past exam paper at the moment. The lecturer has provided us with solutions to the previous years exam paper, not very clear in some cases ...
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0answers
17 views

What is the difference between Gaussian White noise and $iid$ noise and how can I check?

If I understand correctly, a series {$X_t$} is $iid$ noise if there is no trend or seasonal component and the observations {$x_t$} are independent and identically distributed with zero mean, while a ...
0
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1answer
20 views

Inverse of Score and Covariance/Variance matrix

I need to proof the following: $(A'A + B^{-1})^{-1}A' = BA'(ABA' + I)^{-1}$ Where B $\in{R}^{k \times k}$ is a variance matrix, $A\in R^{n \times k}$ of full rank. Unfortunately, I seem to get ...
2
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1answer
20 views

covariance of random variables

Suppose X, Y, W are independent random variables such that X ∼ GAM(2,3), Y ∼ N(1,4) and W ∼ BIN(10,1/4). Let U = 2X − 3Y and V = Y − W . Find cov(U, V ). I know that cov(U, V) = E(U, V) - E(U)E(V). ...
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1answer
32 views

Bias and variance of estimator

I have the following estimator, $E = 1/\bar{X}$ of $E = 1/\lambda$ where X is exponentially distributed with parameter $\lambda$. I'm trying to find the bias and variance of this estimator. For the ...