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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When does Mean Square Error increase?

As far i know, we want the model to include as few regressors as possible because the variance of the prediction $\hat y$ increases as the number of regressor increases. But from the hald cement ...
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26 views

Computing the probability of two variables of the same sample.

Problem: The mean of a variety of apple is 400g with a standard deviation of 50g. If we choose 2 random apples of this variety, what would be the probability that the first one weights 150g more than ...
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32 views

How to use Chi-Square for a rolling die

I'm watching this video. The guy said that this kind of test allows us to check whether the experiment occurs by chance or something wrong with the way the experiment is being done. I have done very ...
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16 views

What will happen if we adopted 1 x IQR to determine an outlier?

What's the percentage of all values that would be considered outliers if we adopted the 1 x IQR rule?
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10 views

Calculating average with special summary statistics

We want to compute the mean of a data set D. The data set is not accessible. Instead we repeatedly gain access to average and size of a subset of D, which contains a data point d $\in$ D and some ...
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30 views

How to measure the similarity or divergence of two distributions with different supports?

Suppose $X$ and $Y$ are two random variables with the distributions $F_X$ and $F_Y$ on the same support $\Theta$.Then KL divergence $D_{KL}(X||Y)$ is a way to measure the statistical distance between ...
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84 views

Why John Tukey set 1.5 IQR to detect outliers instead of 1 or 2?

To define outliers, why we cannot use: Lower Limit: Q1-1xIQR Upper Limit: Q3+1xIQR OR Lower Limit: Q1-2xIQR Upper Limit: Q3+2xIQR
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20 views

Whats the reason of log-odds?

I'm following a course of bayesian inference. Out of the middle they are talking about log odds. For example, say that we have a uniform prior distribution, $\theta$ ~Beta(1,1). $$g(\theta) = ...
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69 views

Poisson Process

Assume emails arrive in your inbox modeled by a Poisson process with a mean of 4 per hour. Let $X$ denote the the time unitl your next e-mail, which is therefore exponentially distributed. Find the ...
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27 views

Value in between given percentiles

If I have a set of data such as 3500,4500,5700,7100 that represent the 25th, 50th, 75th, the 90th percentile how would I go about finding a percentile in between two of the given points? For instance, ...
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21 views

Please clarify what a sample is in statistics

Is a sample 1) DATA from a SUBSET of a population? or 2) A SUBSET of a population? I am not sure if its the Data or the actual Population that the data is derived from.
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Binomial Distribution/ Law of large numbers

I currently have the problem to establish that $ \underset{n \rightarrow \infty}{\lim} \sum_{k=0}^{\lfloor n\cdot l\rfloor} \binom{n}{k} (s)^k(1-s)^{n-k} = \begin{cases} 0, & \text{wenn}~l< ...
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67 views

Is it possible to generate a uniformly distributed random 128-bit number from multiple uniformly distributed random numbers of size <= 32 bits?

If I have a uniformly distributed random number generator of up to 32 bits in length, can I generate a uniformly distributed 128 bit number by rolling my 32-bit random number generator multiple times ...
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103 views

Basic Statistics question - Qualitative or Quantitative and which type of graph

We're doing an assignment based off this data about solar activity. Please correct me if anything I say sounds incorrect. I will upvote all comments. Any help is appreciated! Instructor provided ...
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57 views

how to calculate the marginal distribution of probabilistic principal component analysis

In the book Pattern recognition and machine learning from Bishop equation 12.33 states: $\mathbf{x} = \mathbf{W} \mathbf{z} + \boldsymbol\mu + \boldsymbol\epsilon$ Here $\mathbf{z}$ has a normal ...
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1answer
42 views

How to calculate continous variance of given set of functions?

imagine we have a set of continuous functions over time, which are defined for certain time spans: $$ Q_1(t) = 10 \times t \quad for \quad0<t<5$$ $$ Q_2(t) = 4 \times t \quad for ...
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1answer
28 views

Statistics with probable outcomes

Suppose an instructor ha a class with 14 freshmen, 15 sophmores, 8 juniors, and 10 seniors. Say 8 names will be drawn from a hot to go on a trip. How many different draws of 8 names are there overall? ...
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25 views

The Joint pdf of random variables

Let the joint density of X and Y be given by $f(x,y)= 24xy$ for $0< x < 1, 0 < y < 1, 0 < x+y <1$ and zero otherwise. Are X and Y independent? Let us compute the marginals. For $x ...
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133 views

Maximum Likelihood Estimator where x>= theta

Let X = X1, X2, . . . , Xn be i.i.d. random variables from a population with a density if x ≥ θ f(x;θ) = 2θ^2/x^3, if x ≥ θ, 0 elsewhere. What is the Maximum Likelihood Estimator of θ? Is it unbiased? ...
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30 views

Chebychev's Inequality Consistent Estimator

Let $Y_1,\ldots,Y_n$ be a random sample from the Uniform distribution on [$0$,$\theta$]. We know that $\hat{\theta}_n$ = $\frac{n}{n+1}$$Ymax$ is unbiased for $\theta$. Use Chebyshev's Inequality to ...
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41 views

Joint probability density word problem

If X is the proportion of persons who will respond to one kind of mail order soliciation, Y is the proportion of person who will respond to another kind of mail order solicitation and the joint ...
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14 views

Estimate size of frequency table

I have a frequency table of the top 500 elements, e.g. category1 215,903 category2 152,393 category3 135,404 Additionally, I know the total count of all ...
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18 views

Let $X_1$ and $X_2$ be independent $n(0,1)$ random variables. Find the pdf of $(X_1-X_2)^2/2$.

I understand that $(X_1-X_2)/\sqrt2)$ ~ $n(0,1)$ since it is a linear combination of $X_1 $ and $X_2$ and hence $(X_1-X_2)^2/2$ ~ $\chi^2_1$. I'm having trouble on how to prove/show this ...
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28 views

$Var(\bar{X})$ for a random sample from Bernoulli Distribution

Let $X_1,...,X_n$ be a random sample of size $n$ from a Bernoulli distribution with parameter $p$ where $0< p< 1$ is unkown. (a) Find $\theta^2=Var(\bar{X}).$ (b) Find the value of $c$ so that ...
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24 views

Determining Likelihood Function

Radioactive particles are emitted randomly over time from a source at an average rate of x per second. In n time periods of varying lengths t1, t2,...., tn (seconds), the numbers of particles emitted ...
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23 views

Discrete distributions; find E(X)

For each of the following distributions, find: $\mu=\mathsf E(X)$ $\mathsf E[X(X-1)]$ and $σ^2=\mathsf E[X(X-1)]+\mathsf E(X)-\mu^2$: a) $f(x)= \frac{3! }{ x!(3-x)! }(\frac 1 4)^x (\frac 3 ...
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2answers
123 views

Union of three independent events

Let $ A_1, A_2, A_3 $ be independent events with probabilities $ \frac 12,\frac13,\frac14,$ respectively. How to compute for $P(A_1\cup A_2 \cup A_3).$ My solution starts from using the probability ...
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1answer
28 views

Statistics question regarding mean of exponential distribution

So I have a question in front of me which reads: 'If component lifetime is exponentially distributed with parameter,obtain an expression for the proportion of components whose lifetime exceeds the ...
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60 views

Convergence of probability

So I am getting ready for my first econometrics exam, and we have a lot of these plim (probability limits). Looking at the definition, I have that as n goes to infinity, ...
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20 views

The critical value table

how can I derive the number of critical value table?!Is there any specific formula ?I have searched allover the internet but I didn't mange to find any.
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29 views

Show Y is location-scale if $\sigma > 0$ is unknown

Let X be a random variable having the gamma distribution with shape parameter $\alpha$ and scale parameter $\gamma$, where $\alpha$ is known and $\gamma$ is unknown. Let $Y= \sigma $ log $X$. Show ...
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75 views

Probit regression model: Construction of weighted least squares algorithm

I'm posting a difficult general linear model question which I would like to solve. Question: Consider a probit regression model for $y \in ${$0,1$}:$E(y|x)=\Phi(x'b)$, where $\Phi$ is the standard ...
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2answers
66 views

Having some trouble understanding Probability Distributions

The question is : If X has the Distribution Function F(x) = { 0 for x< -1 , 1/4 for -1 <=x<1 , 1/2 for 1<= x <3 , 3/4 for 3<= x < 5 , 1 for x>=5 } find a) P(X<=3) ...
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1answer
66 views

Comparing MSE of two estimators

Let $\hat{\theta}_1$ be an estimator of $\theta$, which is known to lie in the interval $(a,b)$. Define an estimator of $\theta$, $\hat{\theta}_2$, by $\hat{\theta}_2=\hat{\theta}_1$ if ...
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48 views

Unweighted Mean of Percentages

I have come across an interesting problem, which I'm sure there must be a standard solution to, but lots of Googling hasn't yielded much (which may mean I'm well off track...). I am trying to ...
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49 views

Soft question: Distribution of the kth powers of normal random variables.

If $X_1,..,X_n$ are standard normal random variables then it is knows that: $\underset{i=1}{\overset{n}{\sum}} X_i$ is a normal random vairable and $\underset{i=1}{\overset{n}{\sum}} X_i^2$ is a ...
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14 views

Is auto-correlation a particular and simple form of pattern in time series?

In a time serie context and considering the sign of the variation of the variable auto-correlation means that for couple of measures at (time t, time t+1) the ...
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1answer
19 views

covariance matrix in bivariate distribution

I struggle to understand how exactly you get the covariance matrix in a bivariate normal distribution. The reason is probably that I have no idea how to obtain it at all. In the exercise I have I ...
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1answer
50 views

Powers of random variables always well-defined?

Given the random variable $X$, is $X^{0}$ a random variable? Can we take the expectation $E(X^{0})$? Is $X^{0}$ or its expectation defined or undefined under any conditions (say, on the sample space ...
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1answer
62 views

number of ways to seat 6 students (3 females and 3 males)

we have 3 female and 3 male students traveling to spain. On the plane we have 3 seats on each side of the gangway, kinda like this: |A B C | |D E F| a) Suppose At this point that one female sits to ...
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25 views

Continuity of a CDF: what is wrong about my reasoning?

I want to prove that a distribution function $F$ is continuous at all points in its domain. By definition, $F$ is right-continuous, so it is enough to prove that it is left-continuous. Basically we ...
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1answer
34 views

Finding the CDF using the definition of CDF.

I am not sure how to proceed from here using strictly the definition of the CDF: Let a point be selected from the sample space $\mathcal{C} = \{c : 0 < c < 10\}$. Let $C \subseteq ...
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1answer
17 views

Area of a one-dimensional set.

Let C be a one-dimensional set and let $\;Q(C)=\int_C e^{-x}\;dx\;$. If $\;C_1=\{x\;:0 \le \;x \le 1\}\;$ and $\;C_2=\{x\;:1 \lt \;x \le 3\}\;$ Find the $\;Q(C_1\cup C_2)\;$. My solution: ...
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69 views

conitional probability-probability this given this

According to U.S. Department of Labor employment reports,in 2010, 39.3% of jobs were classified as management, professional, and related occupations (Mgt), 14.5% were Service occupations (Svc), 23.2% ...
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60 views

Econometrics Question

Can someone please help explain this (or provide a website link). I know the answer is (a). To decide whether or not the slope coefficient is large or small, a) you should analyze the economic ...
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161 views

Combination with selective boundaries

Im in a statistics course and I am honestly quite stuck on this problem. I really would love some guidance! "A shelf contains 9 different fiction books and 6 different nonfiction books. A woman ...
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35 views

Probability that the uniform random variable is in Cantor set

Question : Let $U$ be a standard uniform random variable on $[0,1]$, and $C$ be the standard cantor set. find $P(U \in C)$. Is it correct to say that since $P(U = x : x \in [a,b]$) = $1/b-a$. ...
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41 views

Knowing that the weight corresponding to the third quartile […] calculate the value of μ and the value of σ.

The weight in kg, of cubs in a year follow a standard deviation with a mean of μ and standard deviation σ. (a) Knowing that the weight corresponding to the third quartile is 21,3kg and the weight ...
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1answer
108 views

Find the probability that X is within one standard deviation of the mean

The problem asked me to find the mean of poisson distribution, after the mean has been found it asks me the following: "Find the probability that X is at least within one standard deviation of the ...
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19 views

Predicted value of polynomial regression models

Suppose that we have a polynomial linear regression as following $$ Y_i = \beta_0 + \beta_1 X_{i} + \beta_2X_{i}^2 + \epsilon_i, \quad i=1,\ldots, n $$ with $\epsilon_i \sim N(0,\sigma^2)$ and ...