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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How to I use the standard normal table to get the following Z value?

So I am given that $P(X \le 31.5) = 0.05$ and according to the textbook, after standardizing and using the standard normal table we get $$(31.5 - \text{mean})/(\text{standard deviation}) = ...
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1answer
60 views

sample variance of regular polygon upon superimposition of vertices

Given, the vertices of a regular polygon, the centroid here would be the sample mean of the vertices and we assume it to be at the origin. The distance from each vertex to centroid is ...
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24 views

Sampling distribution for sample mean and sample variance in case of Normal population [closed]

Let $X_1,X_2...,X_n$ be a random sample drawn from $N(\mu,\sigma^2)$ popultion. Further let $\bar{X}$ and $S^2$ be sample mean and sample variance respectively. Then Find the joint PDF of ...
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1answer
51 views

Probability of being the millionth customer (What Would You Do?)

I saw this episode of "What Would You Do?" a few months ago, and I keep wondering what would statistically be the best thing to do in this situation. Here is the problem formulation: You are waiting ...
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2answers
39 views

How do I rate smoothness of discretely sampled data? (Picture!!!)

In the sense that the following curves pictured in order will be rated 98%, 80%, 40%, 5% smooth approximating by eye. My ideas: (1) If the curves all follow some general shape like a polynomial ...
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10 views

How close is $\operatorname{argmax}_p E[\log(f(p,\alpha)]$ to $\operatorname{argmax}_p \log(E[f(p,\alpha)])$?

Here $\alpha$ is a random variable and the expectation is taken with respect to that variable. I am wondering if it's the same in any case or there's a theorem quantifying how close both things are. ...
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1answer
19 views

Isolate Sigma and FWHM from gaussian

I have the Gaussian: $a e^{-b^2 (x-c)^2}$ And need to isolate the Sigma and FWHM from it. I believe that $b = \frac{1}{\sigma^2}$ and $FWHM = 2.354(\sigma/2)$ However, I need to program this ...
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1answer
43 views

Current applications of the central limit theorem for binomial distributions

The central limit theorem in the binomal distribution case, also known as the De Moivre–Laplace theorem was historically used to approximate the binomal distribution with the normal distribution. I ...
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1answer
23 views

find the probability that the amount of milk produced is between 45.9 gallons and 50.3 gallons. [closed]

the amount of milk produced each day by a herd of cows is uniformly distributed between 44.6 gallons and 58.8 gallons. find the probability that the amount of milk produced is between 45.9 gallons ...
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0answers
26 views

How is the Variance of this estimator equal to $\theta$?

Currently going through solutions of a worksheet and I don't understand the jump between two lines of working. "$\hat{\theta}_1$ and $\hat{\theta}_2$ are independent unbiased estimators for an ...
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1answer
20 views

proportion confidence intervals

Here is a question from my introductory statistics course. What is the sample size necessary for a two-sided $90\%$ CI for the population proportion $p$ to have a width of $0.2$ when no prior ...
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39 views

% Relation of the Total

I have two factories Production Target Factory A 121.58 126.41 Factory B 110.62 106.45 If ...
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31 views

Expected value of discrete uniform variable

I have a question regarding linear combinations/transformations in statistics. I'm quite sure the answer is relatively easy, but I can't seem to find a solution that corresponds to my solution manual. ...
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0answers
20 views

Derivative of a correlation function

From a big set of data I create a correlation function between a response parameter and three input parameters $(P_1, P_2, P_3)$. $Response = K_1 + K_2 \cdot P_1 + K_3 \cdot P_2 + K_4 \cdot P_3 + ...
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2answers
49 views

If we know X is a Poisson binomial random variable what can we say about mX?

Suppose that X is sum of m independent Bernoulli random variables that are not necessarily identically distributed, and thus it has Poisson binomial distribution. Is mX also a Poisson binomial random ...
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0answers
40 views

PCA “Compact Trick”

This is my first time asking a question on here, so forgive me if I broke any etiquette! My question is related to a question asked here. To paraphrase what is mentioned in the link, consider a ...
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1answer
40 views

Computing $E[ {\rm Tr}\{(ZZ^T)^2 \}]$ for $Z$ Gaussian.

Let $Z \in \mathbb{R}^n$ be a Gaussian random vector with zero mean and $Cov(Z)=I$ where $I$ is identity matrix. How to compute \begin{align*} E[ {\rm Tr}\{(ZZ^T)^2 \}] \end{align*} I know that ...
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0answers
25 views

How do I find middle range with normal distribution?

I am currentLy reading my textbook there is an example that I cant understand. Can someone explain to me the hint given to solve it? Q. Verbal SAT scores follow normal(430,100) distribution. What is ...
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2answers
18 views

Statistics and data analysis

What is meant by confidence interval in data analysis e.g. 95% confidence interval? How does p<0.05 estimate significant difference?
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1answer
51 views

1 scratch off, 1 ticket or both? [closed]

I have a problem that I would like some input on. Free beer(or soda) for best answer. I have several scratch offs and several tickets options. I have the maximum loss(item price) and I have the ...
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1answer
23 views

Check if the weak law of large numbers holds true for the following sequence of random variables

Suppose we have $n$ independent discrete random variables, whose distribution is as follows: $X(k)$, where $k$ is any integer from $1$ to $n$, can take any of three values: $-\sqrt{k}$ with a ...
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17 views

distribution function from a sample

I have a sample about relative error of a measuring device and I wanna know the error probability of that device. What non parametric test must I use to know its error probability distribution of the ...
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1answer
32 views

What is the probability density function of **the multiplication of Gaussian variables**?

Assuming $x_1,x_2,\ldots, x_n$ are $n$ independent variables from standard Gaussian distribution $N(0,1)$. Then we construct a new variable by $y=\Pi_{i=1}^n x_i$. Can anyone show the probability ...
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1answer
40 views

Combining normal distrubutions

I am not sure of the terminology here, if this is a product, summation, or average. How can you take a two unimodal normal distributions and combine them into a bimodal distribution? And then combine ...
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56 views

5 girls and 3 boys are arranged randomly in a row. Find the probability that…

5 girls and 3 boys are arranged randomly in a row. Find the probability that: a) the 5 girls are next to each other,= 2/28 b) the 3 boys are next to each other,=3/28 c) there is one boy on each end, = ...
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1answer
47 views

Which math class can I take to learn how to derive statistical models

I have taken several stats classes and. Have seen many models in action like the normal, poisson, dirchet, etc. and seen several inference tests in action like chisq, ttest and anova. However I'm ...
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2answers
36 views

Difference between two proportions in a Confidence Interval

Ten engineering schools in the United States were surveyed. The sample contained $250$ electrical engineers, $80$ being women; $175$ chemical engineers, $40$ being women. Compute a $90\%$ confidence ...
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1answer
40 views

Monopoly Game Statistics

I was playing a game of monopoly the other day, and in the course of strategizing I came up with the idea that how 'safe' you were in the game was a matter of what your expected income/outcome was as ...
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31 views

How to prove: $E(|tr(x^Tww^Ty)|^k)\leq \|yx^T\|_2^k E(tr(x^Tww^Tx)^k)$?

How to prove: $$E(|tr(x^Tww^Ty)|^k)\leq \|yx^T\|_2^k E(tr(x^Tww^Tx)^k)$$, where $k$ is a positive integer, $x,y$ are fixed vectors, each entry in $w$ i.i.d. follows from an standard norm ...
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1answer
23 views

Generalization of method of least squares to matrix system (Pseudo inverse?)

A and B are two m$\times$n real matrices with m > n. I need to find X: a real m$\times$ m matrix such that $\| A - X B\|$ is minimized. On thing I'm thinking about is using the singular value ...
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1answer
15 views

The sample distribution (pdf) of the sample mean retrieved from gamma distribution

Is it true that the sample distribution (pdf) of the mean where sample is of size n retrieved from a gamma distribution with shape a and scale b is given by ...
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12 views

DRIFT MATRIX in Ornstein Uhlenbeck Process

The Weiner Process was unable to explain Brownian Motion and then there was the need of Ornstein-Uhlenbeck Process. The Ornstein-Uhlenbeck Process describes the Brownian Motion in the presence of ...
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2answers
139 views

Why are these following variance and expected value computations legitimate?

I spent over an hour of my exam's given time to calculate the variances and expected values as given here: Let $p,q\in (0,1)$. The number of costumers entering a supermarket is a r.v. $X$ with ...
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1answer
34 views

About 'Marcinkiewicz–Zygmund inequality'

Marcinkiewicz–Zygmund inequality gives gives relations between moments of a collection of independent random variables. The statement of this inequality can be seen in Wiki ...
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2answers
28 views

how manys ways are there if the order is taken into account?

Three candidates are selected from a certain number of interviewess. if the order is not taken into account, the number of ways the candidates can be chosen is 35. how manys ways are there if the ...
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2answers
35 views

Sampling distribution of $Y = \frac{\ln U_1}{\ln U_1 + \ln (1 - U_2)}$, where $U_i \sim U(0,1), \forall i$

For this problem I have used the fact, $-2 \ln U \sim \chi^2_{(2)}$. But I have doubt on the independence of numerator and the denominator which are $\ln U_1$ and $\ln U_1 + \ln (1 - U_2)$. If they ...
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1answer
41 views

$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}}$ , where $X \sim N(0,1)$

Prove that: $$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}},$$ where $X \sim N(0,1)$ and $c>0$, $t \in\mathbb R$. The problem should be solved easily by using the equality: $$P(X \geq c) = P(e^{Xt} ...
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3answers
50 views

Probability: bakery distributes pies

I'm working through a mathematical statistics textbook, and I can't get a question right. It is a follow-up to this question: At the end of the day, a bakery gives everything that is unsold to ...
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0answers
25 views

How to distribute a cost in a normal distribution

I need to spread out a number so that it reflects a normal distribution. For example, I have an item that cost $500,000$ dollars in year $2050$ and I would like to spread it across with a standard ...
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1answer
41 views

probability of threshold crossing [closed]

Let $\{X_i\}$, $i=1,2 \ldots$ be $\textit{i.i.d.}$ positive random variable distributed as (some) $F(\cdot)$ with finite mean. Let $S_n= X_1+ X_2+ \ldots+X_n$ be the sum of $n$ $X's$ and let $a>0$ ...
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0answers
19 views

Bounds on Chi Squared Distribution

Consider the following hypothesis test: $\mathbf{X} =(X_1,\cdots, X_k) \sim $Multinomial$(n,\mathbf{p})$ and $H_0 : \mathbf{p} = \mathbf{p}_0 = (p_1,\cdots, p_k)$. I know to test this, we construct ...
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0answers
15 views

Indicator variables and instrumental variables

Consider that we have a problem of endogeneity in the classical linear regression model $\operatorname{cov}(x,u)\neq0$. We find an instrument for this endogenous variable. Suppose the instrument is ...
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20 views

Instrumental Variables and orthogonality conditions

To obtain the method of moment estimate for instrument variables, we use the moment condition $z'\varepsilon=0$ in the exact identified case (number of endogenous variables = number of instruments) or ...
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26 views

Leave K out cross validation shortcut

For splines and linear regressions there is this handy shortcut: Let $\hat{f}$ be a spline estimate of a true function $f$, and let $\hat{f}_i^{[-i]}$ be the model fitted to all data except $y_i$. ...
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35 views

With the constraint $E(X)=0,E(X^2)=1$, is Rademacher (symmetric Bernoulli) variable X the best choice to minimize $E(X^4)$?

Rademacher variable $X$ means that $X$ can be either $-1$ or $1$ with equal probability $0.5$. Then my question is that: Is Rademacher (i.e. symmetric Bernoulli) variable X the best choice to ...
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0answers
15 views

Measure of how well a 3D function represents experimental data

I have experimental data of a one dimensional heat equation and corresponding values for a predicted temperatures. Is there any method in which I could statistically analyse the data to determine if ...
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1answer
32 views

Estimating the number of classes in a finite population [closed]

Suppose I have N smarties, each of which is one of C distinct colours. Suppose further that N is known and largish (10,000) but C is not, and that for each colour C there are $c_i$ smarties of that ...
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2answers
34 views

Statistical Significance of a Simple Test

Please help me with this basic question on statistics: If a standard brick is dropped on a standard raw chicken egg from 1 meter; the egg breaks. How many times does this dropped-brick-onto-egg need ...
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1answer
34 views

How to find CDF and PDF of $Y = 4X(1-X)$, given $X\in[0,1]$ [closed]

Let $X$ be uniform on $[0,1]$ and $Y = 4X(1− X)$. Find the CDF and PDF of $Y$.
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Find probability using geometric distribution

I wanted someone to check the solution of this problem (see Rice's book, problem 2.14) Two boys play basketball in the following way. They take turns shooting and stop when a basket is made. Player A ...