# 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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### Consistency of bootstrap estimator that is continuously $\rho_\infty$-Frechet differentiable

Theorem. Suppose $T$ is continuously $\rho_r$-Frechet differentiable at $F$ with the influence function satisfying $0<E[\phi_F(X_1)]^2<\infty$ and that $\int F(x)[1-F(x)]^{r/2}dx<\infty$. ...
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### What's the proper error metric for a 'guessed' result when comparing with the observation data?

Hi Statisticians/Mathematicians, I am trying to compute the error of a 'guessed' value against some observation data, and am wondering what's the right process/approach. For example, there is a group ...
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1 vote
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### Probability of Different Outcomes Being First Given Base Probabilities in a Random Order

I'm given a set of probabilities p1, p2, ..., pn. corresponding to different outcomes. They are then randomly ordered and tested until one returns true or all are false. I want to find the probability ...
1 vote
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### Is every method of moments estimator (MME) asymptotically normal distributed?

As already written in the question, I am asking myself whether every method of moments estimator (MME) is asymptotically normal distributed? Formulated differently, is every (central or non-central) ...
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### Calculate the posterior distribution

How can I solve the letter (a)? Discrete sample spaces: suppose there are N cable cars in San Francisco, numbered sequentially from $1$ to $N$. You see a cable car at random; it is numbered $203$. You ...
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### Bayesian Inference Intractability

When looking at Bayesian posteriors $$p(z \mid x) = \frac{p(x \mid z)p(z)}{\int p(x \mid z')p(z')dz'}$$ The denominator commonly intractable. I understand this is due to the possibility of high ...
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1 vote
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### Question about the null fraction parametrization for a poisson model (Mathematical statistics)

Im trying to understand the following in my mathematical statistics textbook: For a family of Poisson distributions it can be parametrized through its mean \begin{align*} E_{\lambda}(X)=\sum_{x=0}^{\...
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### What is "implied standard deviation from sample size", and how do I calculate it?

I'm doing a research on 538's Pollster ranking and their methodology, and trying to replicate it on another database. However in the process of doing that, I couldn't find the 'expected error', error ...
1 vote
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### Joint probability table not adding up to 1

I got this question asked in my exam today: Suppose there is a group of 9 students that belong to a university. 4 of them study economics, 3 of them study business and the rest study engineer. Suppose ...
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### Get Conditional Density from Joint Density in Abstract Setting

Let $X_1$ and $X_2$ be real random variables defined on a common probability space $(\Omega, F, P)$. Let $\nu$ be a $\sigma$-finite measure on $\mathbb{R}^2$. We suppose that $X=(X_1,X_2)$ has joint ...
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### Linear regression: $\hat{u}$ vs. $u$

I am studying statistics as a major and got some confusion watching $\hat{u}$ and $u$. My professor said $\hat{u}$ is called a residual and $u$ is an unobservable error term. Therefore this is what I ...
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### Deriving the CAPM pricing kernel from the general SDF and consumption-based kernel

I'm reading the paper "Quality minus junk" by Asness et al. (2019) and trying to understand the pricing kernel definition they provide on page 6. The authors present the following pricing ...
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### Calculating the expected value for a complicated pdf

I am given the following cdf (cumulative distribution function) for the random variable $T$ with support $x\in(0, \theta)$. $$F_{T}(x)=\Bigg(\frac{x-\frac{1}{n}}{\theta}\Bigg)^{n}$$ I am asked to ...
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### Stalker Prob Stats [duplicate]

There’s a new family with two children in the neighbourhood, since you’re a stalker and a statistician, you have found out that at least one of the children is a boy. a) Find the probability that both ...
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### Are sample mean and variance unique estimators?

I get that the sample mean and sample variance are unbiased and consistent estimators. But i'm wondering if they are unique estimators that are unbiased and consistent. i.e. for any statistic A, if A ...
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### Tricky Bar Graph or Histogram [closed]

I am trying to determine if this is a bar graph or a histogram. It's the distribution of American families by income. This has been such a struggle for me, but I'm sure there's a simple explanation. ...
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1 vote
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### Law of Total Probability with Extra Conditioning Examples?

I cannot understand the logic behind the Total Law of Probability with Extra Conditioning. I am hoping to get some simple examples that explains it. The ones I have seen so far are still very ...
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I know that we must use the Multinomial Theorem. But why is $$n_1=n_2=n_3=n_4=1?$$ Is it because each teacher is distinguishable and can only be counted once? What does each $$n_1, n_2, ... ,n_r$$ ...