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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Hypothesis testing for variance when the mean is not known

The sugar content of the syrup in canned peaches is normally distributed. A random sample of n = 10 cans yields a sample standard deviation of S(10) = 4.8 milligrams. Suppose that the variance is ...
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15 views

Numerical stochastic ODE: finding E[X(t)] etc at discrete points

So if this is trivial, please tell me I'm an idiot. I guess, given a continuous time stochastic process, one can numerically approximate it at discrete points. Nothing too extreme. I'm new to this ...
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35 views

Joint moment generating function problem

X and Y have the following joint moment generating function: $M_{X,Y}(a,b) = \Large \frac{4}{5}[\frac{1}{(1-a)(1-b)}+\frac{1}{(2-a)(2-b)}]$ Find E(XY) I have gone through this problem several times,...
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116 views

Statistics, least square method

I am having problems with an exercise. I have some observations of the random variable $Y$: $0.17, 0.06, 1.76, 3.41, 11.68, 1.86, 1.27, 0.00, 0.04,$ and $2.10$. I know that $Y = X^2$ and that $X \sim ...
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30 views

Bernstein-type inequality for heavy-tailed random variables

It is known that for independent sub-exponential random variables, the following Bernstein-type inequality holds: \begin{align} \mathbb{P}\biggl(\biggl| \sum_{i=1}^N a_i X_i\biggr| >t \biggr) \...
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11 views

Computing preference using mean and variance?

I have 10 Items, which are available for me to interact with. Interaction with them yields me some reward.I have an expectation/mean/probability and variance/uncertainty of the reward for each item. I ...
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15 views

Finding probability given a sample from 36 with a given distribution

Let the sample mean and variance be based on a random sample of size 36 from $N(4, 121)$ distribution. Find $P(0 < X < 8, 40 < S^2 < 160)$. Since they are independent it would be $P(0<...
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47 views

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}_i) = 0$

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}) = 0$ in the image below (third and fourth line of the proof!). Why?
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8 views

Advice on picking the best 3-set of ingredients based on their rating against different recipes

Statistics are as easy to get wrong as crypto so I thought I'd ask here rather than stackoverflow. I have some data as follows: a bunch of Xs (let's call them ingredients) are being rated by a bunch ...
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40 views

Conditional expectation in terms of CDFs

I have a variable $\omega$ that is lognormal, that is, when we take the log of $\omega$ the distribution is normal. I saw in a paper the following result: $$ E\left[\omega | \omega\geq \bar{\omega} \...
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45 views

Find expected value of discrete trials same probability.

I am struggling with a question. I have found the probability of success to be $\frac{k!}{k^k}$. How do I find the expected number of trials before a success? For example, this is similar to if I ...
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62 views

Probability of earnings from lottery

Question: A city's lottery works in the following way: An individual selects 6 numbers from the first 30 numbers. The city then selects 6 numbers from the first 30 numbers. If the individual selects ...
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103 views

Probability of waiting time

Question: At a railroad junction, a car and a truck arrive between 7:15 and 7:30. A train stops the traffic for five minutes from 7:20. What is the probability that the car and truck waited for ...
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28 views

Finding second moment without standard deviation

Just want to make sure I'm doing this correctly Question: The weight limit of a scale is 5.25 kilograms. W is normally distributed with mean weight 5.15 kilograms and standard deviation $\sigma$. ...
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16 views

About regression model and assumptions

I have the following general regression model $$y=E_{Y|X}[y|x]+u.....(1)$$ Where $u$ is understood as the error. In the basic model there is a common basic assumption about avoid endogeneity, i.e. $$...
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80 views

Find the pdf of $Y = g(X)$, where $X$ is a uniform random variable

The question is as follows: Let $X$ be a uniform random variable over $(-1,2)$. Let $g(x) = |x|$. Find the pdf of $Y = g(X)$. And here is my take so far: $$f(x) = \begin{cases} 1/2 & \text{ ...
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15 views

Writing power series for $AR(2)$ model polynomials

So I have found the following problem in my textbook without solutions, which presents the $AR(2)$ process defined by $$X_{t} = 0.5X_{t-1} + 0.25 X_{t-2} + Z_{t}$$ I am asked what the polynomial $\...
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56 views

Transformation of variables for a non-monotonic function

Question: Let $U \sim \mathrm{Unif}(−α, α)$ follow the uniform distribution on the interval $(−α, α)$ for some parameter $α > 0$ and consider the transformed random variable $X = \sin(U)$. ...
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32 views

Defined $X,Y,Z,W$ find $P(W=0)$ and $P(W=X)$

Suppose $X,Y,Z$ are iid binary random variables satisfying $P(X=0)=P(Y=0)=P(Z=0)=0.5$ and define a new random variable $W$ as $W=X$ if $Z=0$ and $W=Y$ if $Z=1$. Then find $P(W=0)$ and $P(W=X)$. I ...
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419 views

Sheldon Ross vs My TA, what answer is wrong?

I have the solution of this problem, 1) The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is chosen randomly and the object of the game is to guess the chosen three. In one ...
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Describing a linear predictor for a stationary process with a set of equations

In some exam review I received the following question Assume that $\{X_{t}\}$ is a stationary time series, with mean $\mu$ and acvf $\gamma(h)$. Let $P_{n}X_{n+h}$ be the optimal (in terms of ...
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25 views

Independence of $A$ and $\cos B$ [duplicate]

Quick question - if $A$ and $B$ are independent continuous random variables, does that imply $A$ and $\cos B$ are also independent?
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44 views

Calculate sample size given Conditional Probability and level of significance

On the basis of a pilot study, it was found that the predation probability for dark-coloured moths on a dark background is 0.10, in contrast to 0.90 on a light background. What should be the sample ...
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53 views

Probabilty Mass Function for runs scored in an over

The probabilities of scoring 0, 1, 2, 4, and 6 runs on a given delivery in a cricket match are 0.4, 0.35, 0.15, 0.075 and 0.025 respectively. Assuming that the probabilities of a wicket, a wide ball ...
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35 views

an urn containing 5 green and 2 red balls

An urn contains 5 green and 2 red balls. One ball is drawn at random and its colour is recorded. This selected ball is then replaced in the urn and 3 more balls of the same colour are added to the urn....
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62 views

How to Compute Var($4X-Y$) on this case?

Let $X$ be the number of $1$'s and $Y$ be the number of $2$'s that occur in $n$ rolls of a fair die. $(a)$ Compute Cov($X; Y$ ). Solution are given in other post: Compute Cov(X,Y) while X is ...
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38 views

Probability of average people coming to work

I am working on this: A restaurant employs $9$ people. $2$ bartenders, $3$ waiters and $4$ work in the kitchen. It has been observed that on any day the probability of an employee to call in ...
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21 views

Finding distribution of sample variance from $\chi^2$-distribution

I'm looking at the sample variance, $$ s^2 = \frac1N \sum_i^N (x_i - \mu)^2 \equiv \frac{\sigma^2}{N} \chi^2 $$ For normally-distributed samples, I know that $$ p(\chi^2) = \frac{1}{2^{N/2}}\frac{1}{...
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39 views

testing correlation coefficient in a bivariate normal distribution

How can I show that $\dfrac{\hat{\rho } \sqrt{N-2}}{\sqrt{1-\hat{\rho}^2}}$ has a t-student distribution with $N-2$ degrees of freedom. I think I have to write it as a quotient of a normal $(0,1)$ ...
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Prove $\mathsf E^Y \mathsf E^Y [X] = \mathsf E^Y [X]$

Prove that$$\mathsf E^Y \mathsf E^Y [X] = \mathsf E^Y [X]$$ by using $$\mathsf E^{Y=y} [X] = \int x\,p(x\mid y)\operatorname d x$$ or $$E^{Y=y} [X] =\sum_x x\,p(x\mid y)$$ I am having trouble on ...
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35 views

The four assumptions on linear regression

It is clear that the four assumptions of a linear regression model are: Linearity, Independence of error, Homoscedasticity and Normality of error distribution. My question is does any of these four ...
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128 views

Distribution of quotient of random variables

If $X$ and $Y$ are independent random variables such that $X\sim \Gamma(a,b)$ and $Y\sim\Gamma(a,c)$. What is the distribution of random variable $\frac{Y}{X+Y}$? Any help with this ?
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36 views

Estimators in the case of refression with normally distributed errors

How can it be shown that the Maximum Likelihood Estimator and the Least Squares Estimator are equvalent in the case regression with normally distributed errors? Any help will be appreciated! Thanks in ...
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Probability of a difference dependent samples, normal distribution but unknow parameters

I have two samples, $X_0, Y_0$, containing the marks in a test before and after training course over the same workers, thus dependent paired samples. X, Y can be considered Normal, with the same ...
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36 views

Maximum Likelihood Estimator of $\theta$

I have the following question I tried to answer I got answer that same like this answer Is this true answer? (Note that: in the question $0<p<\frac{1}{2}$, but in this answer $...
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12 views

Find the error (cumulative function of abs.cont. random variable)

Let $X$ be an abs. cont. random variable. Then, for $k \in \mathbb{R}$, $prob(X \leqslant k \leqslant \alpha X)=prob (X\leqslant k)-prob(X\leqslant \frac{k}{\alpha})$. I cannot understeand why it ...
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16 views

Using normal distribution to create confidence interval

Let $Y~N(\mu, 1)$. Use the fact that $P(\left | Y-\mu \right | < 1.96\sigma) \approx.95$ to construct an interval $(a(Y), b(Y))$, such that the probability $\mu$ is in the interval is approximately ...
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19 views

finding the likelihood function for a sample

I need some help with finding the likelihood (or log-likelihood) function for this problem: A sample $x_1,...x_n$is generated as follows: for $i=1...n$ 5-bit binary string $w \in \{s_0,...s_{31}\}$ is ...
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31 views

random variables and their maximum expected value

I read a statement in a book and I can't realize what exactly it means, and why it is true. a set of random variables are given. in general, the expected value of the maximum of random variables ...
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43 views

What is the Edgeworth Expansion of the binomial distribution?

For a standardized binomial distributed random variable $\tilde B_n$ we have $$P(\tilde B_n\le x) = \Phi(x) + \frac {q-p}{6\sqrt{npq}} (1-x^2) \phi(x) + \frac{R_1\left(np+x\sqrt{npq}\right)}{\sqrt{...
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81 views

Delta method for Chi-squared distribution

I am not familiar with chi-squared distribution and also delta method, please help me to clarify. Given $Z_n$ ~ $\mathcal{X^2}$. The mean and variance of $Z_n$ are n and 2n respectively. The ...
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Is Standard Deviation Independent of Sample values

I have a rule to implement which says that if standard deviation of a set of measurements is > x, then y is true. Problem is that my values could be anything like: 1,1.1,1.2,1.2,1.3.. or 100, 100....
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27 views

Fisher information for a single sampling of an exponential distribution

I am viewing an example of finding the Fisher information for a single sampling from an exponential distribution where: $$P(x|\theta) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$$ The score $S$ is $S(x|\...
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37 views

Make x the subject of a double exponential equation

Many statistical packages allow a double exponential function to be fit to your data (below), yielding five constants A, B, C, R and S, which can be used to describe the curve. $$ y = A + B * R^x + C ...
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80 views

Probabillity of failures involving exponential distribution

I have come across this problem: A room is lit by $10$ light bulbs. The lifetime, X, of the light bulbs follows an exponential distribution with mean $ \mu$ = $1000$ hours. In a time window of $...
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What does ${50}\choose{4}$ mean in statistics?

I have a test tomorrow in statistics and was wondering what the following means? $$\binom{50}{4}$$ My professor along with most of my classmates have a calculator they can just plug that into. The ...
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49 views

Understanding the Normal Distribution?

If a sample is normal with observations independent and identically distributed: $\mu|\sigma^2 \propto N(\beta \,,\,\sigma^2/\, n_0)$ How can I show that $\mu\,|\,x_1,x_2,....x_n\,,\,\sigma^2 \sim ...
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42 views

joint pdf of X and Y

the joint pdf of X and Y is defined as: \[ f_{X,Y}(x,y) \begin{cases} \frac{3}{2}, & \text{if } 0\leq x \leq 1 \text{,}x\leq y \leq 1 \\ \frac{1}{2}, & \text{if } 0\leq x \leq 1 \text{,}0\...
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29 views

Estimating parameter $\alpha$ from $n$ independent observations.

For a distribution $f(x|\alpha) =\frac {1 + \alpha x}2 $ we have that an electron is moving forward $(x \lt 0)$ or backward $(x \gt 0)$. How can I estimate $\alpha$ from $n$ independent observations ...
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rate of convergence central limit theorem

Let $X_{1},X_{2,\ldots }$ be i.i.d. with finite second moment random variables. With mean $\mu$ and $\sigma^{2}$ Then the classical CLT states: $\sqrt{n}(\overline{X_{n}}-\mu)$ convergences to $N(0,\...