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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Why is the mean of the $t_1$ distribution undefined? [duplicate]

The pdf of the $t_1$ distribution is symmetric about $0$, just like the pdfs of $t_k$ for $k>1$, so why does it not also have an expected value of 0?
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1answer
48 views

approximating a uniformly distributed random variable

Suppose that $U$ is a uniformly distributed (continuous) random variable on $[0,1]$. Let's say that I am interested in finding 3 discrete points $u_1,u_2,u_3$ which approximate $U$ in some sense. My ...
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1answer
26 views

t value question

You are conducting a study to see if students do better when they study all at once or in intervals. One group of 12 participants took a test after studying for one hour continuously. The other group ...
0
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1answer
23 views

Find approximation for size of population over time

Assume you start with a population of an objet of size $1$. Assume that a new objet of size $1$ is born at each date and that existing objects double in size in each period. Over time the sequence of ...
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1answer
16 views

Figuring out weighted criteria numbers/percentages?

I'm a bit confused on how I should go about accomplishing this/unsure of the terminology. So let's say I have a list of $10000$ voters. I want to get a composition that looks like: $34%$ Democrats ...
2
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1answer
44 views

sampling distribution question

Need clarification on a binomial sample example: we drew a sample of size $100$ from a binomial($m = 2$,$p = 0.2$) distribution and observed $76$ of the $x_i = 0$, $20$ of the $x_i = 1$ and $4$ of ...
0
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1answer
31 views

A machine produces memory sticks of varying lengths…

Full question: A machine produces memory sticks of varying lengths, distributed uniformly between 2 and 12 mm. Memory sticks longer than 10 mm do not meet the design criterion and must be scrapped. ...
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0answers
21 views

Moment generating function trouble understanding solution

Suppose that Y is a single observation from an exponential distribution with mean θ. Use the method of moment-generating functions to show that $\frac{2Y}\theta$ is a pivotal quantity and has a ...
4
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1answer
41 views

Average volume of set of cubes using the mean and variance of its side lengths.

First, i tried this question: The side lengths of a set of squares have an average of 5 and variance of 4. What is their average area? Let X = The Side Length From this question , I knew we had to ...
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0answers
18 views

Confused with this $G$ to find the maximum likelihood estimation!

Let $p_{ij}$ denote the probability that an observation falls into cell $(i,j)$ of a two-way contingency table containing $r$ rows and $c$ columns. A total of $x_{oo}$ observations are taken and ...
-1
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0answers
5 views

Second Order Conditions and Maximum Likelihood Estimator for Normal and Exponential distributions

I can't seem to show the Second Order Condition for the MLE of the exponential distribution is <0. Does anyone have any hints? Same problem for the normal distribution when looking for MLE of the ...
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0answers
29 views

How do I find sample size for testing means in statistics?

Is there a reason why they altered the original formula by multiplying the population variance by 2?
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20 views

Shatter coefficient and VC dimension of a grid in $R^d$

Given $\epsilon>0$, partition the cube $[0, 1]^d$ with square of side length $\epsilon$. The total number of square in the partition is $$ N = \left(\frac{1}{\epsilon}\right)^d. $$ What is the ...
1
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1answer
17 views

How to find $\Phi^{-1}(\beta)$

I need to find $\Phi^{-1}(\beta)$ when $\beta=0.1$ (or any number but for example) but I'm not quite sure how to find it using the normal table inversely like this. I've tried googling and looking ...
1
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1answer
18 views

using Negative values for a box plot (box whisker)

I am drawing a box plot for a question where the data set is the bulb life time(in hours) My 5 number summary is ...
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 ...
0
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1answer
27 views

What area of statistics deals with such kind of problems?

Consider $2$ samples from the starting normal distribution with parameters $\mu=0, \sigma = 1$ with size $N$. Find the variance of the random variable $\xi$ equal to average sum of $1$st sample - ...
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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
24 views
+100

Sufficient sample size for 1 out of X sets

I want to create a "game" that aims at supporting the decision making. The user has to consecutively select one out of two random items. Each item belongs to one of X sets. My question is how many ...
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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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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 ...
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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?
5
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1answer
70 views

Can the partial sums of independent random variables with no normalization converge in distribution to a constant?

If $\{X_n , n\ge 1 \}$ is a sequence of independent random variables and $X_n$ is nondegenerate for at least one $n\ge1$, can there exist a finite constant c such that $S_n = \sum_{j=1}^n X_j ...
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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 ...
0
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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 ...
2
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3answers
31 views

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 ...
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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 ...
4
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1answer
69 views

Why do we like sticking random variables into their own distributions?

Let $X$ be a random variable taking values in the set $S$. It has some distribution $f(s)$. Often in statistics, we are interested in the real valued random variable $f(X)$. Here are some examples: ...
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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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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 $$ F^{-1}_X(x) = \sum_{j=0}^{N} s_j^{(X)} \psi(\xi)$$ ...
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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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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 ...
2
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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
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 ...
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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 $p(v)$ ...
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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, ...
0
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1answer
18 views

Question about an R script obtained online (for a $5$-card poker hand simulation)

I got this R script from the following online source: ...
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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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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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
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1answer
32 views

Markov Process graphical representation

I don't understand how the picture has been constructed. Specifically how $\mu^{11}=-(\mu^{12}+\mu^{13}+\mu^{14})$ and $\mu^{44}=-\mu^{43}$ has been graphically represented. Here $\mu^{ij}$ is the ...
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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 ...
0
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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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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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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. ...
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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 ...