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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Sample variance equation

I'm studying statictics, I don't understand why the equation like this i attached. x bar is sample variance https://www.youtube.com/watch?v=D1hgiAla3KI : 5:08
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5 views

How to show sufficient statistics are complete

By writing out the likelyhood function, I can show that $(X_{(1)}, X_{(n)})$ is sufficient statistics, but how to show they are complete?
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5 views

Confidence interval for an overfitted model may be wider.

(a) Consider the multiple linear regression model: $Y = X_1β_1 + X_2β_2 + \epsilon$, where Y is n × 1, $X_1$ is $n × p_1$, $β_1$ is $p_1 × 1$, $X_2$ is $n × p_2, β_2$ is $p_2 × 1$, and $\epsilon$ ...
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0answers
6 views

Multiple imputation for fuzzy logic

In my data set I have missing values for all variables and one particular variable has 12 out of 25 data points missing. So I used multiple imputation method to handle the missing values. What I want ...
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1answer
8 views

Modulus of Z (Normal distribution)

The random variable $Z$ is distributed such that $Z \sim N(0,1)$ find the probability of $P(\left|Z\right| >2.4)$. How to solve this modulus type of question ?
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10 views

Random Variables kith order statistic

If a random variable has density f(x) = αc^(α)x^(−α−1), x > c, where c and α are positive constants, we say that it has a P(c,α) distribution. Consider a random sample of n observations on a P (1, 3) ...
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1answer
30 views

Cats and Dogs = Idenpedent events

I did not get this question. Could you explain it to me? In a building for 24 apartments. It is known that there is only one dog in 8 apartments and a single cat in 6 apartments. How many apartments ...
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14 views

Probability Data Management [on hold]

A bag contains 54 black marbles and 63 white marbles. Use Pascal’s Triangle to determine how many combinations and how many permutations are possible if 7 marbles are drawn out of the bag.
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2answers
29 views

T distribution with n degrees degrees of freedom

I would like to prove that $\displaystyle \frac{\bar{X}\,\sqrt{n}\,}{\hat σ^2}\sim t_{n}$. Note that x~N(0,$σ^2$) and they are iid. Could someone explain why $\displaystyle ...
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1answer
14 views

inequality probability between order statistics of two independent distribution

Suppose we have two independent distributions $F_1$ and $F_2$ and from each distribution, we draw $k$ variables. Let us represent the $k$ i.i.d. variables from $F_1$ as $\{X_1, X_2, \ldots, X_k\}$. ...
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1answer
20 views

Standard deviation of the product of gaussians

What is the standard deviation of the product of two random variables that each have Gaussian Distributions? I don't even know where to begin on this problem.
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1answer
77 views

I'm taking a statistics class right now, and I get stuck on these problems. [on hold]

For a random variable $W$ where $P(W = 0) = 0.1$ and $P(W = 1) = 0.2$ and the density of $W$ for values between $0$ and $1$ is $f(w) = 1.4w$, draw a graph of the CDF. Is this a valid probability ...
2
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1answer
20 views

Statistical bias and the probability of an outcome.

A town referendum has occurred. The question posed to voters was YES or NO on a local law. There were 3 methods of voting: Electronic machine (voting booths), absentee ballot, and affidavit ballot. ...
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1answer
13 views

Derivation of t(n-1) distribution

While trying to prove that $\displaystyle \frac{\bar{X}\,-\,\mu}{S/\sqrt{n}}\sim t_{n-1}$ I came across a manipulation that I can not seem to understand the reasoning behind it. Why does ...
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2answers
25 views

Why is uncertainty in mean less

Any measurement, say length of any object, will have some errors. The random errors that are present in the measurement can be reduced if we take mean of a large number of samples. This is because the ...
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1answer
23 views

calculate median for following data [on hold]

class interval F 45-49 14 ...
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1answer
28 views

Calculating conditional probability of discrete uniform r.v.

X is a discrete uniform random variable on $\{a, a+1, a+2, ... , b\}$ with mean 7 and variance 4. Find $Pr[X \leq 6| X > 4]$ I'm not familiar with the discrete uniform distribution. I was ...
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3answers
48 views

Statistics (unsure how to do it)

A person's resting heart rate is the lowest number of heart beats per minute when fully relaxed and without distractions. Age, fitness, genetics, health status and gender affect the resting heart ...
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0answers
24 views

Not sure what formula to use? (what to solve for?)

The question states, "The weight of people in a certain pacific island is normally distributed with a mean of 175 lb. and a standard deviation of 33 lb. They want to design a one-person canoe that ...
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1answer
18 views

Application of Law of Large numbers (1)

If we have an i.i.d random variable $X_i$ with mean variance $(\mu, \sigma^2)$. By Law of Large number, we have $\bar{X}\rightarrow^p \mu$. But can we use Law of large number as well and claim that ...
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10 views

Cumulative Distribution Function Proof with Discrete Random Variables [on hold]

Let $Y$ be a discrete random variable on the natural numbers $0,\, 1,\, 2,\, 3,\, 4,\ldots$ Let $F_y$ its CDF. Prove that $E[Y] = \sum_{k=0}^\infty(1-F_y(k))$
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28 views

Sampling substrings of a beaded necklace to determine the necklace composition

I have a necklace composed of 100 beads, where each bead is one of 13 colors. If I am only able to look at one 4 bead sub-sequence at a time (connected, as they would be on the necklace) , how many ...
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0answers
8 views

Repeated Measures ANOVA [on hold]

What is the model equation for one-way repeated measures ANOVA? Is that for the 2-way rANOVA similar to the 2-factor experiment with interaction term present in the model equation as follows: Yijk = ...
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1answer
19 views

Wilcoxon signed-rank test

While reading Wikipedia, and my teacher's notes I found that Wilcoxon signed rank test for $n>10$ is given like below: Under null hypothesis, W follows a specific distribution with no simple ...
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0answers
18 views

Divergence based robust inference

I have learnt that the inference based on minimizing the following divergence is robust to outlying observations for some specific range of $\alpha\in\mathbb{R}$. $$D_{\alpha}(g,f) = ...
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13 views

Why are the Real and Imaginary parts of a field in k-space uncorrelated?

I am in the process of generating a (real) Gaussian random field $\delta(\vec{x})$ from a given power spectrum $P(k)$. The way I define the power spectrum is, in Fourier space, $\left\langle ...
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1answer
30 views

Determining bounds for change sum of continuous r.v.'s

I'm trying to understand how to determine the bounds when computing the sum of continuous random variables. Here is a sample question: X and Y have the following joint pdf: $f_{X,Y}(x,y) = 4xy, 0 ...
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34 views

Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. Link here If this is not acceptable - administrators ...
4
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1answer
28 views

sufficient statistics of a sequence of normal random variable

If $X_1, X_2\ldots,X_n$ are independent variables with $X_i \sim \mathcal N(i\theta,1)$, $\theta$ is an unknown parameter. What is a one dimensional sufficient statistic $T$ of this sample? I have a ...
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0answers
13 views

What course would you recomment to take? Elementary Statistical Methods or Introduction to Biostatistics? [on hold]

I am required to take either or for my major (Nursing). What course would you recommend? Which one is easier? What are the pros and cons about choosing one over the other. What exactly are each course ...
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2answers
31 views

If $x_i$ is from a random sample is $Var(\bar x \mid x_i)=0$?

If $x_i$ is from a random sample, is the conditional variance of the mean (or the sum of squares, really any statistic based on $x$) just treated as a constant? I saw this in a OLS variance of a ...
5
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1answer
41 views

Why do polynomial regressions have larger variance at the end?

In reading the book "An Introduction to Statistical Learning with Applications in R", I came across this graph: It shows that the point-wise variance is larger at the ends of the regression curve. ...
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1answer
34 views

3-sigma approximation

I am making a system involving a sensor who has to be really precise. I found on their datasheet a diagram that shows the typical performance of the sensor. There's the mean value, the +3 sigma, ...
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3answers
60 views

Doubt about a probability excercise

I'm a statistics teacher at a college. One day a student came with a doubt about an exercise about probability. The text goes like this: A person has two boxes $A$ and $B$. In the first one has ...
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1answer
22 views

Double integral proving that a function is a probability density

If $$g(x,y)=f(x+y)/(x+y)$$ for $x,y>0$ and $$\int_0^{\infty} f(z) \, dz = 1$$ How do you show that $$\int_0^{\infty} \int_0^{\infty} \frac{f(x+y)}{x+y} dx \, dy = 1$$ as well?
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what is the intuition behind the SRSWOR formula?

I earlier asked about Slovin's Formula, and learned shortly thereafter that it was derived from this formula. $n=\dfrac{n_0}{1+\dfrac{n_0}{N}}$, Where $n_0=\dfrac{z^2p(1-p)}{e^2}$. So, breaking it ...
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3answers
52 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
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11 views

SSE distribution in simple linear regression

I'm looking at the typical simple linear regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, where there $\epsilon_i$s are iid $N(0, \sigma^2)$ random variables. We have unbiased estimates ...
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2answers
27 views

defective component and probability

An electronics industry uses three plants from A, B and C in the ratio of $3$ to $2$ to $1$. However $1\%$ of components manufactured by A, $2\%$ of components manufactured by B and $3\%$ of ...
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2answers
28 views

Games and statistics

Three individuals A, B and C alternate in contention of a game according to the following rules: A plays with B and the winner plays with C. The game continues until one of the individuals to win two ...
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0answers
5 views

Concentration bounds on Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...
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1answer
9 views

Binomial Distribution formula

If $X\sim B(n,p)$, show that $P(X=r+1)=P(X=r) \cdot \frac{p(n-r)}{q(r+1)}$ for $r=0,1,...,n-1$ My attempt, $P(X=r+1)={_n}C_{r+1}(p)^{r+1}(1-p)^{n-(r+1)}$ How to proceed then?
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17 views

Calculate pdf and cdf of exponential random variable

I am not looking for an answer as this is for a homework exercise i just want to be pointed in the right direction so i can learn how to do it. The question is: Let X ∼ Exp(λ). Calculate the cdf and ...
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23 views

How many poker hands until statistically significant winner

How many poker hands do I have to play to determine a statistically significant winner? What is the best approach to get a 95% confidence interval? To give some more context: I have been building a ...
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1answer
16 views

Summatory problem | Ordinary least square estimator

How I can transform the first expression in the second? \begin{align} \hat{\beta}_{1} & =\frac{n\sum X_{i}Y_{i}-\sum X_{i}\sum Y_{i}}{n\sum X_{i}^{2}-\left(\sum X_{i}\right)^{2}} \\ & = ...
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1answer
42 views

Rumor and probability

31 people in a community, a person has a rumor to a second which, in turn, repeats to the third, etc. At each step the person receiving the rumor is randomly chosen among 30 people available. a) Find ...
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2answers
39 views

Two urns, A and B, each with two drawers.

If you have two urns, A and B, each with two drawers. The urn A has a gold coin in a drawer and a silver coin in the other drawer while the urn B has a gold coin in each drawer. An urn is chosen at ...
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20 views

Wick's theorem: Classical Version. Derivation question.

I am trying to prove the classical version of Wick's theorem: For a set of random variables ${a_i}$, with covariance matrix $M$ and $\rho(\vec a)$ a Gaussian probabilitiy density: $$\langle a_j a_k ...
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4 views

assumptions of t-test of correlated terms [closed]

Good Day ,i just want to ask if what are the assumptions of t-test of correlated means.I've been browsing the internet since yesterday because im just so curious if what does this statisticall tool ...
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0answers
30 views

Uniformly boundedness of convolutions

Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$. Let $X_1,\dots,X_n$ ...