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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Convergence in distribution to a standard normal [duplicate]

How to show $$ \lim_{n\to\infty}\left(1-\frac{t^2}{2n}-\frac{t^3}{2n^{\frac32}}\right)^{-n}=e^{\Large\frac{t^2}{2}}\,\,? $$ LHS is expanded and approximated form of a moment generating function. RHS ...
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Normalizing the Second Moment of $n$ Discs.

Consider $n$ non-overlapping discs of diameter $d$ positioned (centred) at $P_1,\dots,P_n$ ($\|P_i - P_j\|\geq d, i\neq j$). Graham and Sloane use the second moment as a measure of compactness for ...
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25 views

Probability and Statistics random independent variables

I can't figure out how to determine if these variables are independent. Any help would be greatly appreciated. Random variables x and y are described by the PDF: $$f(x,y) = \begin{cases} k,& ...
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33 views

Compute mean and variance

I have as much numbers as the computer you are sitting now can store in main memory, assume a collection N. I want to compute the variance and the mean of them. I do that by applying Knuth's ...
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48 views

I need help with this review question. Business Calculus/Statistics

The question is: The scores on a test have a mean of 100 and a standard deviation of 8. A personnel manager wishes to select the top 60% of applicants. Find the cutoff score. Assume the variable is ...
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31 views

Need help with math and statistics. [duplicate]

The world population is ~7 billion Social Classes: Top .001% is ~70,000 Second .01% is ~700,000 Next .1% is ~7 million Rest 99.9% is ~6,993,000,000 (billion) The odds of 1 person in the ...
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49 views

Question about probability? World Population - (Question Revised)

The world population is ~7 billion Social Classes: x: Top .001% is ~70,000 y: Second .01% is ~700,000 z: Next .1% is ~7 million r: Rest 99.9% is ~6,993,000,000 (billion) p: population ...
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42 views

Probability error

I perform $N$ independent trials with $M$ successes. The probability of success is therefore $P=M/N$. Can I assign a sample-size-dependent error to the probability based only on this information? i.e. ...
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32 views

Probability of a specified sequence in a random data set

This is a problem which I have encountered while programming, but I imagine this community would be better able to solve it. Suppose we have a number, N, of boxes in a row. In each of these boxes is ...
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41 views

conjugate prior

A class of sampling distribution is a conjugate family of a prior distribution, if the posterior distribution belongs to the same family for all priors and all samples. Why is this phrase incorrect?
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distribution of $X^2 + Y^2$

Suppose $X$ and $Y$ are independent uniform distributions between $(0,1)$. What is the distribution of $X^2 + Y^2$? I derived that the pdf of $X^2$ is $\frac{1}{2\sqrt{x}}$ for $0\leq x \leq 1$. How ...
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Narrowing down a location on a grid based on multiple data points

I'm working on a program for triangulating wireless device locations on a map. So far I've cooked up the triangulation algorithm, but the problem is that wireless signals can bounce around depending ...
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74 views

Finding Survival Function given hazard rate

Let X be a random variable defined for 0 < x < 4 with hazard rate $$lambda(t)=1/(4-t)$$ for 0 <= t <= 4. find the survival function, S(x) = P(X>x). Using the formula S(x) e^-integral from ...
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What is ${\rm cov}(e_i, \hat y_i)$ in simple linear regression?

The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ What is ${\rm cov}(e_i, \hat y_i)$? What is ${\rm cov}(\epsilon_i, \hat \beta_1)$? What is ${\rm cov}(e_i, \epsilon_i)$? For 1, I am writing ...
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33 views

Calcuate the test statistic

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
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33 views

Uniform distribution parameter estimation

How do I estimate the parameter, when the distribution is $\mathrm{U}(-a,a)$? I get $a=\min(X_i)$, $b=\max(X_i)$, when it's $\mathrm{U}(a,b)$, but these do not work here.
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64 views

The critical value

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
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52 views

The Objectivity of Statistical Testing

I have a very generic question about applied statistics. Suppose, to make things simple, we have a biased coin with probability $p$ of landing heads. We want to determine if our coin is truly fair - ...
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45 views

Calculating the critical value

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
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1answer
80 views

Calculate Critical Value

Experience in investigating insurance claims shows that the average cost to process a claim is approximately normally distributed with a mean of 80 dollars. New cost-cutting measures were started and ...
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1answer
260 views

Confidence interval multiplication

The question looks pretty simple but I can't get my hands on it: Say I have a probability which is the product of two other independent probabilities $p = p_1p_2$. I have estimated each probability ...
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1answer
44 views

Hypothesis testing on minimum of exponentially distributed random variables

I am completely stuck with the following problem, because I do not know how to start: Let $X_1,...,X_n$ be independent and exponentially distributed with unknown parameter , and let ...
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53 views

Source needed: Does asymptotic normality yield asymptotic unbiasedness and consistency?

Assume that $$\sqrt{n}(\hat g - g(\theta)) \xrightarrow{d} Z, $$ where $Z$ is $N(0,\sigma^2)$. Does this already imply asymptotic unbiasedness and/or consistency, i.e., $$ E[\hat g] \rightarrow ...
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$MLE$ of $\theta$ when $X_1 , X_2 , …, X_n$ is a sample with pdf $f(x, \theta) = e^{\theta - x}; x \ge \theta$

How can we find the $MLE$ of $\theta$ when $X_1 , X_2 , ..., X_n$ is a sample with pdf $f(x, \theta) = e^{\theta - x}; x \ge \theta$ ? $L(\theta) = \prod_{i = 1}^{n} e^{\theta - x_{i}}$ $L(\theta) = ...
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Paired t test, finding mean and variances

Paired t-test: Let x1, x2,...,xn be iid ∼ N(mu1, sigma^2). Let y1, y2,...,yn be iid ∼ N(mu2, sigma^2). Suppose also that the pair of observations within subject i, (xi,yi), has correlation rho. This ...
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How to know when to use which test statistic?

I'm studying for my stats exam and I seem to always get stuck on which test statistic to use. This is even after many examples that I have done. For example: With this question: I attempted to ...
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21 views

How to get F-test p-value

Suppose we can choose from two differents catalysers. 10 substances are taken from the first one and 12 from the other one. $S1 = 0:14$ and $S2 = 0:28$, can we reject at $\alpha = 5%$ the hypothesis ...
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81 views

If I have a random variable X with a given Probability Density Function, How do I find the PDF of the area of a circle with radius X?

To find the PDF of the area of the circle, do I just substitute the PDF of the random variable X in for the radius in the circle area equation?
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36 views

Calculating confidence interval - formula

I have the following problem that I get the feeling I'm mixing formulas. ...
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1answer
21 views

Please explain uniform distribution to me [duplicate]

I am confused about Uniform Distribution why does $$P(v < 2b1)$$ equal 2b1 ?
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79 views

What's the densitiy of the product of two independent Gaussian random variables?

Suppose that $X,Y$ are two scalar independent normal random variables, $X \sim N(\mu_X,\sigma_X^2)$, $Y \sim N(\mu_Y,\sigma_Y^2)$. I'm particularly interested about the case where we don't assume ...
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Game Theory and Uniform Distribution question?

In an Auction , two players are bidding. Their bids will be a unknown fraction of their valuations. The valuations come from a uniform distribution $$[0,1] $$ If Player 2 bids $$ v/2 $$ and Player ...
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22 views

Addition of two box plots

I have two box plots that I'd like to add together to form one box plot representing both plots in one. Here's an example of two box plots: At a school 200 boys and 200 girls participated in a test. ...
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Hironaka 1964 theorem in the context of S. Watanabe 2009 book

I am trying to read the following book of S. Watanabe: "Algebraic Geometry and Statistical Learning Theory". More particularly, I am currently interested in chapter 2 and Hironaka (1964) theorem on ...
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84 views

Card probability

There are two 10-card decks, consisting of 5 red cards and 5 blue cards each. Both are shuffled separately. One card is then dealt from each deck and compared. This is repeated for all 10 pairs of ...
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How to find the MLE of the mean of Gamma distribution

If I parameterize Gamma distribution in the way as $\Gamma(\alpha,\frac{\mu}{\alpha})$, am I able to find the maximum likelihood estimator of $\mu$. Here, $\alpha$ is the shape parameter, ...
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Relationship between Fisher information conditional on MLE and Fisher Information conditional on True Value

As described in most textbook, the formal definition of Fisher Information is a function conditional on true value of parameter $\theta$, i.e. $I(\theta)$. Today I came across this phrase: ...
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41 views

Alternative Hypothesis

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
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1answer
28 views

Self-Selection Bias Intrinsic to Survey Samples?

I'm new to stats so bear with me in asking this question. I'm sure my novice will shine through. I've noticed that with any survey there is an intrinsic opportunity for self-selection bias (There is ...
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1answer
79 views

Null Hypothesis

Experience in investigating insurance claims shows that the average cost to process a claim is approximately normally distributed with a mean of 80 dollars. New cost-cutting measures were started and ...
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1answer
150 views

is UMVUE unique? is the best unbiased estimator unique?

guys Here is the question: is the best unbiased estimator unique? My understanding is that the best unbiased estimator must be the UMVUE, so the original question turns into the uniqueness of UMVUE. ...
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71 views

Normal Ratio Distribution with CDF Method

I think I'm missing something glaringly obvious here that's causing problems for me in the entire subject. I have two independent standard normal random variables, X and Y ~N(0,1), and I need to find ...
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1answer
43 views

Lickly hood estimators and discrete random var

If have discrete random variable X and have the observations that X=1, twice, X=3, once and X=4, six times is the likely hood of these observations, L = P(X=1)^2 . P(X=3)^1 . P(X=4)^6 or L = ...
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34 views

Approximating the expectation of a function with sampling

I'm reading a paper (section 5.1) that approximated the expected value of a function $f(X,Y)$ of two random variables using Gibbs sampling. As far as I know the expectation of $f(X,Y)$ is defined to ...
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1answer
16 views

Variences and adding them from independent random variables?

If I have 3 random varibles X, Y and Z and X=Y+Z then var(X)=var(Y)+var(Z), but Y=X-Z therefore var(Y)=var(X)+var(Z), it is clear that these two contridict, so what makes one of them right and the ...
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Proving a statistic is sufficient and NOT complete

Let X be a single observation from the distribution , $P(X=x)= \begin{cases} \theta, & \text{if $x$ =-1} \\ (1-\theta)^2\theta^x, & \text{if $x =0,1,2,3,...; 0<\theta<1$} \\ ...
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Sufficient statistic for normal distribution

Let X from a Normal distribution $(\theta,1)$. a) Find a sufficient statistic for $\theta$. b)Is $S_n^2$ sufficient for $\theta$ My answer for part a) The joint p.d.f= $1 \over (2\pi)^{n/2}$$e^{{-1 ...
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Estimating parameter of random sample

5 random numbers were recorded: 100,32,76,52,17. If we know that these are elements of random sample took from set {1,2, ..., N}, how do we estimate the parameter N?
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Lack of memory property of probability distributions

According to wikipedia lack of memory property applies to geometric and exponential distributions. I was trying to apply it to binomial distribution. Am I modelling my question correctly? So imagine ...