Tagged Questions

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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Mean of a sampling distribution.

Suppose $\hat{p}=1/\overline{X}$ is an estimator of the parameter $p$ of a population variable $X\sim\text{Geo}(p)$. Suppose $p=0.36$ and $n=25$. What is the mean of the sampling distribution? This ...
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1answer
19 views

Probability - Airplane overselling tickets

Few days ago, I came across a question for probability in one of the interview. Question : The same small commuter plane has 30 seats. The probability that any particular passenger will not ...
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1answer
9 views

Bivariate Continuous Distributions

What is the marginal density of $X$ and $Y$ given the probability density function, ${f(x,y)= \lbrace3x ,\;\;0\le y\le x\le1}$
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25 views

Exam grades and bell curve

What is the mathematical explanation for the tendency of exam grades to conform to a bell curve? Initially, I was thinking it should be explained via the central limit theorem, but it's not clear to ...
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1answer
15 views

Poison distribution variance,probability. and mean.

Let $X$ be the poisson random variable such that $P(X = 2) = 9P(X=4) + 90P(X=6)$ a) find the mean and variance of $X$. b) find P(X $\geq 1$) c) find P(X $\leq 10$) Ok so for the first question I ...
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16 views

Lottery Ticket Probability [on hold]

At a certain retailer, purchases of lottery tickets in the final 10 minutes of sale before a draw follow a Poisson distribution with mean = 15 if the top prize is less than 10,000,000 and follow a ...
2
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0answers
18 views

CI for the expected value of the sum of two dependent normal RVs

Let's consider 2 dependent, normally distributed R.V.s, $X_1$ and $X_2$. The means, $\mu_1$ and $\mu_2$ are known, as well the covariance matrix $\Sigma$. Let's consider the following random ...
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0answers
11 views

Question about regression model

Suppose you fit (estimate the parameters of) a regression model, obtaining $\hat{Y}$, $\hat{B}$, and $\hat{E}$. And you fit a second regression model , using $\hat{Y}$ x matrix from previous model ...
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1answer
19 views

Showing a group of observation is standard normally distributed

Let $X_1,X_2\dots$ be a sequence of independent RVs such that $X_{n}$ is binomial with parameters $2n - 1$ and $1/2$. Define $$Y_{n}=\frac{2(X_{1}+X_{2}+\cdots+X_{n})}{n} -n$$ Show $P[Y_{n}<t]\to ...
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0answers
10 views

Random Sample taken [on hold]

A random sample of 300 people are taken. What is the probability that at least 100 of them are over 180cm in height given average height = 175 and standard deviation = 10?
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1answer
18 views

how to show for a simple regression with an intercept and one independent variable$ R^2 = r ^2$ , where $r$ is the ordinary correlation coefficient.

how to show for a simple regression with an intercept and one independent variable $R^2 = r ^2$, where $r$ is the ordinary correlation coefficient. Here is where I'm at. $R^2= ...
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0answers
17 views

Clarification on proposition in paper

I'm referring to proposition 1, page 309 of [1] The proposition itself reads: Let $\pmb x_1,\ldots,\pmb x_n,\ldots$ be iid with $\pmb x_i=\pmb B\pmb u_i+\pmb t_0$ where $\pmb u_i$ has ...
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2answers
29 views

Can 2 different random variables have the same CDF?

I'm looking for proof that two different random variables can have the same Cumulative Distribution Function; in other words, I'd like to disprove that a CDF uniquely defines a random variable. ...
0
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2answers
23 views

Co-relation Coefficient

$X$ and $Y$ are jointly continuous random variables. Their probability density function is: $$f(x,y) = \begin{cases}2x & \mbox{if } x\in [0,1], y\in[0,1] \\ 0 & \mbox{ otherwise ...
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1answer
16 views

Finding the probability of a randomly selected event?

I know I'm over-thinking the following question, I just need to know how to start! In a certain population of women 4% develop symptoms of a classic disease, 20% are smokers, and 3% are smokers and ...
1
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1answer
24 views

Joint probability density function probability

$X$ and $Y$ are jointly continuous random variables. $$f(x,y)=\begin{cases}kx & x\in[0,1], y\in [0,1]\\0 & :\text{otherwise}\end{cases}$$ a) What value of $k$ makes this a density ...
2
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1answer
21 views

Expectation of vector valued functions

Let $t_1,\ldots,t_m$ be $m$ random variables that are independently and identically drawn from a Bernoulli distribution with a constant parameter $p$. Now, we define some functions of ...
2
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1answer
31 views

Find the value of k which makes f a density function.

Observe the following probability density function for a continuous random variable X $$f (x) = \begin{cases} k\sqrt x (1-x) &\text{ for }x\in(0,1)\\ 0 &\text{ otherwise} \end{cases} $$ Find ...
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0answers
13 views

Bayes rule with discrete prior

Assume following discrete prior on $m$ where $m \in \left\{2, 3, 4, 5\right\}$ and $p(m)= .14, .13, .2, .32$ accordingly. If $f(x|m)= \exp[(n/2\sigma^2)(x-m)^2]$. What is the posterior values for ...
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0answers
11 views

Beyes coin problem

assume coin with probability p. Probability is unknown but there are possible values of {.01, .02, .03} with probabilities {.26, .05, .03} 1) if there are observed 12 heads and 13 tails, with 25 ...
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1answer
29 views

Drawing Probability Density Function

Can someone help me to draw this pdf? I really don't have an idea how to convert a function to pdf. Thank you p(x | c) = 1/3 for 1 <= x <= 4 and P(c) = 0.5
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0answers
26 views

Probabilty Models and distribution techniques

Coliform bacteria are distributed randomly and uniformly throughout river water at the average concentration of one per twenty cubic centimeters of water. Part (c) In testing for the concentration ...
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0answers
10 views

How can I minimise the MSE in binomial distribution

$X_1$-Bin(n,p) and $X_2$-Bin(n-$x_1$,p). n is the unknown total population. I am given that $T_b$ is the estimator of n where $T_b$=a$X_1$+b$X_2$. Also, the ...
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0answers
17 views

The Hessian Matrix I calculate is twice as much as it should be. Why?

I have a function "fkt." In this example, let it be as simple as $y=a \cdot x+b$. I have a real dataset with values obeying to the model. After regression of the points to the model, I find the ...
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1answer
20 views

Apples Binomial [on hold]

"Apples are packaged automatically in 3-pound bags. Suppose that 4% of the time the bag of apples weighs less than 3 pounds. If you select bags randomly and weigh them in order to discover one ...
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1answer
29 views

When to use Central Limit Theorem or Cramers Theorem

In for example this paper the authors say The central limit theorem provides an estimate of the probability \begin{align} P\left( \frac{\sum_{i=1}^n X_i - n\mu}{\sigma \sqrt{n}} > x \right) ...
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0answers
15 views

Questions dealing with Poisson distribution technique

"A waste disposal company averages 6.5 spills of toxic waste per month. Assume spills occur randomly at a uniform rate, and independently of each other, with a negligible chance of 2 or more occurring ...
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0answers
31 views

Joint multivariate normal distribution

If $\mathbf{y}$ has a multivariate normal distribution $N_n(\mathbf{a},\sigma^2\mathbf{I})$, can I say that $(\mathbf{y},\mathbf{a}^\intercal\mathbf{y})$ also has a multivariate normal distribution? ...
1
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1answer
24 views

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult?

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult? I am consider taking a undergraduate course in my college called mathematics of statistics and in the ...
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3answers
32 views

What is the correct statistical language to conclude using type II error?

Update: This question arises when I read a neuroscience paper. In the natural science community, people are generally less careful about the correctness of statistical language. I am clearly aware of ...
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1answer
29 views

50% plus or minus 13% is good or bad guessing. How was this reached?

I watched a documentary some years back and I cant find it anymore. It was either about genetics or twins (specifically identical twins). But I recall a test was done to see statistically if identical ...
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3answers
27 views

Suppose U is a uniform random variable over [0, 1] [on hold]

This is one of two HW problems that I'm positing that I have no clue how to go about. Suppose U is a uniform random variable over [0, 1]. a.) Show that $Y = (b - a)U + a$ is uniform over [a, b] b.) ...
2
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1answer
31 views

If $E(W) = \mu$ and $Var(W) = \sigma^2$

This is one of two HW problems that I'm positing that I have no clue how to go about. If $E(W) = \mu$ and $\operatorname{Var}(W) = \sigma^2$ show that $E\left(\frac{W - \mu}{\sigma}\right) = 0$ and ...
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0answers
11 views

Will XORing any data with random data produce a random result?

Provided you have a stream of input data and a stream of random data both in the set (0,1). The random is data truly random, that is, unpredictable by the user and ...
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1answer
24 views

Statıstıc problem

Will I use binomial distribution for this question? Can you help me please thnk you
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0answers
10 views

Equal fraction of explained variance in PCA

Let's suppose we obtained this variance decomposition after performing a PCA: My question is, what can we tell about the original data? I know the variance of each principal component is the ...
0
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1answer
29 views

Probability Mass fuction for scratch ticket

A lottery ticket has 4 squares, each with either a star or an X. The printer printing these tickets has a 20% chance of printing a star per individual square. If the amount paid per star is as ...
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0answers
18 views

Statistic problem

Can you help me to solve this problem pls,I have exam and I am studyıng. What wıll I use, bınomial or Other thing ? Thank you
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1answer
18 views

Problem of Statistics

Let $x$ be a variable assuming values $1,2,\ldots,k$ and let $F(1)=n,\ldots,F(n)$ be the corresponding cumulative frequencies of the 'greater than' type. Show that $$\text{Mean of }x=\frac{ ...
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2answers
21 views

Statistic binomial dist

Can you help me to solve this question pls, I consider that I Will use binomial distrıbutıon but I couldnt
0
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1answer
17 views

Increase variation between 2 values

How can I calculate the percentual increase variation ? June: $ 2.574.724,83 July: $ 4.041.072,22 I want also do the inverse, i.e. if I know the increase ...
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3answers
34 views

Generate random number with specific probability distribution

Well consider I have a uniform random number generator. How would I craft a function which takes as parameter this RNG, such that the distribution is following a given function? Or more to the point: ...
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0answers
8 views

Finding midpoint in class interval to calculate mean for cumulative frequency graph.

In a cumulative frequency graph (or histogram), the data is often given in class intervals. To calculate the estimated mean of the data, the formula is: (SUM m*f)/(SUM f) where m = midpoint of the ...
1
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1answer
53 views

Geometric vs Arithmetic returns differences

Been reading some notes that say when calculating returns, using the geometric methodology (1+returns, performing a division between the two returns and - 1 from the result) for computing returns is ...
0
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0answers
24 views

Survival models and differential equations

I have a question regarding survival models and differential equations. Is it possible to translate survival models ( in survival analysis) into differential equations? For example can we write the ...
2
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2answers
43 views

Convergence Rate of Sample Average Estimator

Let $X_1, X_2,\cdots$ be i.i.d. random variables with $E(X_1) = \mu, Var(X_1) = σ^2> 0$ and let $\bar{X}_n = {X_1 + X_2 + \cdots + X_n \over n}$ be the sample average estimator. Is there a way to ...
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0answers
22 views

About matching.

A population of $n$ people, each having $m$ "features" $f_1,f_2, \dots ,f_m$ (for instance, where they live, brand of milk they consume, annual profit, etc...). Now, let $f_1$ a basic feature. Is it ...
0
votes
2answers
33 views

Given a list of N integers, how to find out if the second derivative is positive or negative?

Let's say I have a list where N=10, such as [45,34,56,22,33,44,34,34,43,35]. I would like to know if the second derivative is positive or negative, in other words, if the rate of change of these ...
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0answers
8 views

Tailored Expectation

Let $x=[x_1,\ldots,x_m]^T$ be a random vector whose elements are drawn iid from some distribution which is parametrized by some parameter $\theta$. (i.e. $x_i\sim P_{\theta}(x_i)$). I want to know if ...
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0answers
8 views

How to model and plot non-stationary mean-valued data

What would be the best way to plot a distribution of the data whose mean is non stationary. For example: if I have a data series, say y = [100, 97.3, 95.2, 93.2, 91.1, 91.2, ... 0] which yields ...