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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Need clarification on a passage. (Autism Prevalence)

Passage verbatim: According to a review by Silverman et al. (2010), the incidence of autistic disorder is 0.6–1.0 percent in the population. The disorder is four times more common in males ...
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12 views

Expected value for a mixture distribution [on hold]

Consider a variable $X$ with pmf $f_1(x)$ with probability $p_1$, and pdf $f_2(x)$, with probability $p_2$, where $p_1 + p_2 = 1$. If $Y = 1$, then $X \sim f_1(x)$, and if $Y = 2$, then $X \sim ...
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1answer
19 views

Conditional expectation of second moment given sum of iid variables.

We have $\xi_i \geq 0$, $\forall i = \overline{1,n}$ (i.i.d. variables). Assume that $S_n = \xi_1 +...+ \xi_n$. It is easy to show that $\mathrm{E} (\xi_1\vert S_n = 1) = \frac{1}{n}$. Now we want ...
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9 views

Number of lists at some Kendall-Tau distance

I'm looking for the number of ranked lists (of length n) that are at a given distance d (0 <= d <= n(n-1)/2) from some list of length n (any list, it doesn't matter from which one). I can't ...
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6 views

R-program CompQuadForm/sum of squares of correlated normal variables.

I am trying to use the R-program to calculated sum of squares of two correlated normal variable with non-zero means. I have used the Imhof and Davies algorithme and they work well if means equals ...
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14 views

Accounting for non-existing values in a population.

I have run into a problem representing some data. Lets say you are doing an chemical experiment 100 times. 80 times you get the data you need so you can compute standard deviation, variance, mean, ...
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27 views

Hypothesis testing - Beta coefficient

Apologies if this is trivial, just linke me somewhere. I'm currently taking statistics 101, I can't wrap my head around the hypothesis testing of coefficients. As follows, the t-test reads $$T=\frac ...
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1answer
25 views

A random sample of size 5 is drawn from the pdf $f_{Y}(y) = 2y, 0\leq y \leq 1$. Calculate $P(Y_{(1)} < 0.6 < Y_{(5)})$. [on hold]

A random sample of size 5 is drawn from the pdf $f_{Y}(y) = 2y, 0\leq y \leq 1$. Calculate $P(Y_{(1)} < 0.6 < Y_{(5)})$. (Hint: Consider the complement.) Attempt: The pdf of the largest order ...
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18 views

Which model to be used for predictive analysis [on hold]

I have a problem where i have been given set of data against month example Month | Data1 | Data2 1---------5--------5 2---------6--------7 Consider the data 1 be the temperature and data 2 be the ...
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1answer
24 views

Joint probability distribution

$Y_1$ and $Y_2$ are jointly distributed with density $f(y_1,y_2)=4y_2^2 \qquad 0 \leq y_1 \leq y_2 \leq 1$ Determine the following: $P$( max {$Y_1,Y_2$} $< 1/2$) $P(Y_1+Y_2 < 1/2)$ ...
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1answer
19 views

Given a pdf $f_{Y}(y)$ and $n$ random observations. Find probability that last observation will be the smallest number in all the sample?

Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the entire sample? attempt: ...
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30 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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2answers
17 views

Probability of the highest order statistic below the population median.

What is the probability that the highest order statistic of a random sample of size n from any continuous distribution is below the median ( population median ) of that distribution.
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36 views

Presentation of 2 images in a random but counterbalanced way

Problem: For 18 trials randomly a ‘left’ labeled image or ‘right’ labeled image is shown. The first 9 trials should contain the opposite number of left images as the last 9 (a.k.a. counterbalance). ...
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2answers
23 views

do discrete probability distribution functions need a countable number of outcomes?

Everywhere I see on the internet they say that discrete probability distribution functions have a countable number of outcomes, and continuous have uncountable infinite number of outcomes. However if ...
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12 views

Unbiased Estimator and Variance in Polling

Say a pollster conducted m = 16 polls among people who voted in the 2010 presidential elections, and reports that 55% of the respondents would vote for John Smith. But the pollster did not report how ...
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8 views

Statistics matching answer check please

1.0.0968 < p-value < 0.1056 2.0.2119 < p-value < 0.2266 3.0.0278 < p-value < 0.0316 4.0.3422 < p-value < 0.3682 Possible Answers A. Ha: mu > 2.3, z* = -0.78 B. Ha: mu ...
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8 views

MSE in case of log-transformed dependent variable

Let's consider the following log-linear model: $log(Y_i) = \alpha + X_i\beta + \epsilon_i$ for i = 1, ..., N The fitted value is: $\widehat{log(Y)} = \hat{\alpha} + X\hat{\beta}$ Assuming ...
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1answer
16 views

Hypothesis testing, t procedures [on hold]

A realtor claims the mean income of households in a certain community is \$300,000. To check this claim, a potential resident samples 30 incomes of households in the community, and obtains a mean ...
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23 views

Critical values for hypothesis testing?

How do i determine the level of significance if i know the the critical values, and how do i do the opposite, on a normal distributed curve. I am asking because I am at the moment trying to calculate ...
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10 views

Non parametric estimators for noisy funcions

Suppose there is a function $f(a,b,c,\ldots)$ of $M$ variables (fixed numbers, not random variables). Add some Gaussian noise to this function: $$ g(a,b,c,\ldots) = f(a,b,c,\ldots) + ...
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32 views

A property of the hazard function of the normal distribution

I have a problem that I can't figure out. Define $$\Gamma\left(x\right):=\frac{\phi(x)}{1-\Phi(x)}$$ where $\phi(x)$, $\Phi(x)$ are the density respectively cumulative distribution function of the ...
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One-to-one correspondence between mean value and parameters

I am currently taking a course in statistics, and in this course we are considering linear models $\mu = X\beta$ where $\mu \in L$ and $L = col(X)$ is a linear subspace of $\mathbb{R}^n$, $X$ is the ...
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15 views

What is the limiting distribution of this Markov Chain?

Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$: Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the ...
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CLT, mle, variance [on hold]

This is a practice problem that I don't know how to do. Let X_1,...,X_n be an i.i.d. sample from an exponential distribution with the density function. f(x/T) = (1/τ)*e^(-x/τ) 0<= x <= ...
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3answers
28 views

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement.

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement. Let $X$ be the number of yellow balls on the first two draws and $Y$ the number of ...
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0answers
13 views

Estimate of shared variance for n samples of x and y

I am performing a t-test on n different samples of both $X_1, X_2,...,X_k$ and $Y_1,Y_2,...,Y_k$. To begin with I want to assume that all 2*n samples have the same variance but that they do not have ...
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Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
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1answer
31 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(x=0)$.Here is my solution... $\mu= \frac {p(2) 2!}{p(1)}$. then how can find the mean..thanks
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What is the transformation that maps a Gaussian distribution to a Beta distribution?

Suppose X is a random variable with Gaussian distribution over domain $\mathbb{R} = (-\infty, +\infty)$, with PDF function $f_X$. And Y is a random variable with Beta distribution over domain ...
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1answer
25 views

Probability of Sample Variance Given Variance

I am trying to solve a problem that I have never seen before and cant seem to find a way to solve it so any help or tips would be appreciated! Here's the Problem: Suppose a considerable amount of ...
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23 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
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2answers
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Variance of two functions

I have a problem where Var(X) is given as 8100, Var(Y) is given as 10,000. Var(X+Y) = 20,000. If X is increased by 500, Y is increased by 8%, such that the new formula is X+500 +(1.08)Y. How would I ...
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1answer
12 views

AP Probability problem on independence

This is a in-class practice problem. Suppose that the probability that a person has to park illegally and that he gets a parking ticket is 0.07. Last year Sam recorded data and found that because of ...
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1answer
36 views

Shortcut to finding $E(XY)$

The question says "Find $E(Y|X)$ and hence evaluate $E(Y)$ and $E(XY)$" The joint pdf is $$f_{X,Y}(x,y)=\begin{cases} 8xy, & \text{ for } 0< y< x < 1, \\0, & \text{ elsewhere } ...
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1answer
6 views

Poisson probability statisitics

In Poisson distribution, mean of babies born w/ defect is $1$ per month. What is the probability that exactly $12$ or exactly $14$ babies will be born w/ defect in $6$ months?
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identify nature of missingness for categorical variables

could you please give me some hints for identifying the nature of missingness for categorical variables' missing value? I mean, I gave a fast search on google scholar but I didn't find anything ...
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1answer
16 views

concept of one-tailed hypothesis testing

When we assume that the null hypothesis is true in one-tailed test for mean, we assume that the population mean is equal that value indicated in the hypotheses. Why do we not assume some other value ...
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1answer
17 views

Bivariate GBM - crosscovariance

I have troubles concerning a correlated bivariate GBM with identical drift and diffusion rates. Let $dX^i_t = \mu X^i_t dt + \sigma X^i_tdW^i_t$ and $E[dW_t ^idW^j_t] = \rho_{i,j}dt$ If $X_0^i = ...
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1answer
15 views

Inequality involving different diameter average

I have found an assertion in a scientific book (Hinds, Aerosol Technology, 2nd Edition, 1998, p. 83-84) that claims: Given the general form [here for grouped data] for the diameter of an average ...
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1answer
18 views

Is there an interpretation of the Beta Distribution?

There are cases in probability where one distribution has an "interpretation" in terms of another distribution: X ~ Gamma(k,1/m) for positive integer k, can be interpreted as the distribution of ...
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Recommendation needed in graph theory and statistics to be used in football predictions.

The following is a very simple model of what I am working on. I just need some advice since I don't have graph theory background. Suppose that A played at home against B and won by 3 goal ...
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28 views

Probability that the sum of random variables is less than some value

I would like to obtain the probability that the sum of random variables is smaller than some predefined value. Saying, $X_1, X_2, ..., X_n$ are independent random variables that come from the same ...
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24 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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1answer
12 views

How can I calculate the variance of a minimum of two random variables with different pdf and cdf?

I am trying to solve a problem where I need to find the variance of min (a,b). a is actually a function of a uniformly distributed r.v. while b is another r.v. with pdf and cdf as f and F. The support ...
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1answer
12 views

Bayesian Statistics: Estimators and Posterior Probability

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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7 views

Bayesian Statistics … Γ(α,β) Posterior Probability and Estimators

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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Probability of joint distribution

We were given some exercises to do to prepare for an upcoming quiz and there's one question that I'm struggling on. If $X ∼ N (μ = 10, σ^2 = 4)$ and $Y ∼ N (μ = 8, σ^2 = 16)$. Assume that X and Y ...
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8 views

Confidence interval question in Introduction to theory of statistics [on hold]

Problem: $X$ is a single observation from $$\theta \exp(-\theta x)I(0, \infty)(x)$$ where $\theta > 0$. a. $(X, 2X)$ is a confidence interval for $1/\theta$. What is the confidence coefficient? ...
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2answers
26 views

Finding expected value for random variable $X$ given a joint probability density function $f(x,y)$

I've been given $f(x,y) = 6y$ with boundaries $0 \leq y \leq x \leq 1$. How do I find the expected value of $x$?