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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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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7 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
17 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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0answers
5 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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0answers
6 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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0answers
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
14 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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0answers
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 ...
2
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0answers
5 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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20 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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0answers
8 views

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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0answers
14 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 ...
-1
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0answers
17 views

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
21 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 ...
2
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0answers
10 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 ...
1
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0answers
12 views

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) = ...
1
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0answers
21 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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0answers
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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 ...
0
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1answer
16 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 ...
1
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0answers
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
29 views

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 ...
0
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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
35 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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12 views

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 ...
1
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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 ...
1
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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 ...
1
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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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0answers
4 views

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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0answers
27 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 ...
3
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0answers
20 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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0answers
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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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2answers
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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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0answers
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 $x$?
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2answers
17 views

Want to find the maximum of an unnormalised density function.

Assume $\{Y_i\}$ are iid generated from a gamma distribution with shape $\alpha$ and rate $\beta$, $n$ is the number of $Y_i$. I have an unnormalised density function about $\alpha$ as follow: ...
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2answers
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What is the probability that you will get at least one matching suit in 4 draws without replacement from a standard deck of cards?

I was wondering if someone could help me out with this one. I missed the lecture for this topic and am struggling to catch up. Could someone possibly explain this one to me? Thanks.
2
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1answer
29 views

Find $E(X)$ and $Var(X)$

In a box there are $30$ balls, $20$ are black and $10$ are red. Let $X$ be the number of red in a selection of two balls drawn without replacement then $$X=I_1 + I_2$$ where $I_1 = 1$ if red is drawn ...
0
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1answer
25 views

question on normal distribution [on hold]

A mean of 25 mpg and a standard deviation of 5.8 mpg for highway driving. Assuming that a normal distribution can be applied. a) What gas mileage would put a car in 85th percentile for all cars? b) ...
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I need to compare 6 groups. I can use Kruskal-Walis test. Any other statistical tests I can use?

I have 6 groups of cell cultures I want to compare. I have the data on each of the groups, ex. the viability in each group (90%, 85%, 87%, 78%, 88%, 90%) and so on. How do I compare them? And how do ...
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1answer
24 views

Marginal Probability of Stochastic Process

I have a wide sense stationary stochastic process x(t)=asin(2πf0t)+bcos(2πf0t) where a & b are independent gaussian random variables. How can I find the Marginal probability of x(t)? I am ...
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0answers
21 views

Covariance Matrix Proof - Confusion with Cov(X,X) = Covmat(X)?

I have completed a proof regarding variance, covariance, and the covariance matrix. I think I have made a mistake regarding an assumtion. I need to show that $var(\{f\}) = a^TCovmat(\{x\})a$ Where ...
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1answer
22 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
0
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1answer
54 views

Meaning of symbol $:=$

Can anyone tell me the meaning of this symbol $:=$ I couldn't find it online. It came up while I was studying joint probability of Gaussian random variables.
1
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1answer
25 views

Using CLT to calculate probability

The question I'm trying to answer says that the mean weight of luggage checked by a randomly selected passenger is 40 lb, and standard deviation is 10 lb. Luggage weights are independent. What is the ...
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1answer
25 views

chance to get number 4 in a fair dice in 100 throws [on hold]

I am novice in Stats. If number four has appeared 64 times when a fair die was thrown 540 times? Using the above to get number 4 again in next throw (541st throw) Thanks Rodney
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0answers
11 views

mean & variance relating to bias

I am trying to prove something is biased, and I am given that its a SRS with mean μ = 0, and variance > 0. I think that since variance > 0, it should be biased, but I'm not sure. What does μ = 0 ...