Questions tagged [statistics]
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.
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How can you measure how "shuffled" a deck of cards is?
A few days ago I asked for some methods of measuring how shuffled a deck of cards was. Predictably there were a lot of suggested methods, which got me thinking, which is the best one? I think it'd be ...
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Limiting distribution of binary variable (Central limit theorem fails)
Suppose we have a random variable
$$Y_i = i \text{ with probability } \frac{1}{i}$$ and $0$ otherwise. Here all the $Y_i$ are independent.
We can redefine $X_i = Y_i -1 $ so that $E(X_i)=0$.
Then the ...
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Is there a well-defined `uniform' distribution on $C([0, 1])$?
I'm wondering whether we can define a uniform distribution on the space of continuous functions over a compact set, e.g. $C([0, 1])$. If so, then how should I rigorously describe it? And how can I ...
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The sum of eigenvalues of integral operator $S(f)(x)=\int_{\mathcal{X}} k(x,y)f(y)d\mu(y)$ is given by $\int_{\mathcal{X}} k(x,x) d\mu(x)$?
Setup: Let $(\mathcal{X},d_{\mathcal{X}})$ and $(\mathcal{Y},d_{\mathcal{Y}})$ be two separable metric spaces. Let $M^1(\mathcal{X})$ be the space of Borel probability measures on $\mathcal{X}$ with ...
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Hottest Days of The Year
Recently, there has been much talk in the media of it being the hottest day of the year so far. It has always seemed to me that there are likely many more of these in the northern hemisphere than the ...
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Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$
Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: \...
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Does this calculation have a name, or a generic formulation?
Background Informatiom
I would appreciate help in identifying or explaining this operation:
To calculate each of the $n$ values of $f(\Phi)$:
Sample from the distribution of each of $i$ parameters, $\...
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Only three types of limit of distributions truncated to a finite interval in the upper tail?
Suppose random variable $X$ has a continuous probability distribution with an unbounded upper tail; that is, the CDF of $X$ (call it $F$) is absolutely continuous and $F(x)<1$ for all $x\in\mathbb{...
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Choosing $H_0$ and $H_a$ in hypothesis testing
There seems to be some ambiguity or contradiction in how to correctly choose the null and alternative hypotheses, both online and in my instructor's notes. I'm trying to figure out if this stems ...
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empirical quantile function - uniform convergence
Let $X_1,...,X_n$ denote independent and identically distributed random variables, with $X_i \sim F$, $1 \leq i \leq n$. Assume $F$ is continuous. Then we know that its generalized inverse (quantile ...
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Sum of two independent random variables: distribution function and quantile function
If $X,Y$ are two independent random variables with CDFs $F_X,F_Y$, their sum has CDF $F_X \star F_Y$ ($\star$ is the convolution product).
What can be said about the quantile function of $X+Y$ ? The ...
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How is Optimal Transport algorithmically related to the Assignment Problem?
In optimal transport, we calculate the distance between two probability measures $\mu$ and $\nu$ over the compact set $[a,b]\subset\mathbb R$, using the Earth Movers distance which is a special case ...
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Show that $Cov(\bar{y},\hat{\beta_1})=0$
Show that $Cov(\bar{y},\hat{\beta_1})=0$
For those unfamiliar with statistics, Cov(A,B) refers to the covariance function. $\bar{y}$ refers to the average of the response (dependent variable). $\hat{\...
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Sum of best X dice in Y dice rolled (or roll X pick best Y) odds/calculation
Background: In many pen and paper RPGs there is often an option or bonus/penalty to rolls that incorporates rolling multiples of the required die and taking the best or worst of those rolls for your ...
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How to get the general form of the solution of exercise 5.4-2 of CLRS as showed in wikipedia?
Exercise
Suppose that we toss balls into b bins until some bin contains two balls. Each toss is independent, and each ball is equally likely to end up in any bin. What is the expected number of ball ...
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