I am trying to understand some basic ideas of econometrics (and mathematical statistics) from the precise, mathematical point of view, avoiding vague explanations. I am beginning to learn about these notions from the introductory, undergraduate level, but I have a good mathematical background. Could you please explain precisely some basic notions?   What I mean is the concept of a random sample, first. If there are some mistakes or other inaccuracy, correct it, please.  Suppose we have some statistical population. Precisely, a population is a probability space $X$ plus some random variable, say $\xi$. We may imagine that $X$ is a set of people living in our city and $\xi$ maps each person to his or her height. It is okay, isn't it? Well, what is the random sample of size $n$? I understand it just intuitively. What about the precise definition? "A random sample of size $n$ is a set of $n$ random variables, which are well distributed..." - I read in Wikipedia and many other sources. That is what is very strange. Suppose $\theta$ is some random variable defined on $X$. How can I build the corresponding sample of size $1$? I think that the sample of size $1$ is just a point of $X$, without any random variables. Is it possible to build such a correspondence generally? I know that my question was raised here many times in different forms, but I'd like to reformulate it, trying to get the underlying principles of statistics without the loss of mathematical rigour. So, could you tell me if there are precise definitions in statistics?
10160
What is the difference between a population and a sample? What common variables and statistics are used for each one, and how do those relate to each other?
74541
I'm running an experiment where I'm gathering (independent) samples in parallel, I compute the variance of each group of samples and now I want to combine then all to find the total variance of all the samples.   I'm having a hard time finding a derivation for this as I'm not sure of terminology. I think of it as a partition of one RV.   So I want to find $Var(X)$ from $Var(X_1)$, $Var(X_2)$, ..., and $Var(X_n)$, where $X$ = $[X_1, X_2, \dots, X_n]$.  EDIT: The partitions are not the same size/cardinality, but the sum of the partition sizes equal the number of samples in the overall sample set.  EDIT 2: There is a formula for a , but it only covers the case of a partition into two sets, not $n$ sets.
51403
Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that:  \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} &amp; \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 &amp; \mbox{otherwise}. \end{cases} \end{equation}   Define $Y = ZX$. Show that $Y \sim N(0,1)$. Determine $\operatorname{Cor}(X,Y)$. Are $X$ and $Y$ independent?   Progress:  I'm basically stuck at the beginning. I got as far as $P(ZX &lt; y) = P(Z &lt; y/X)$, which I'm not even sure is correct. I'm also not clear on what it means by defining $Y = ZX$ - does it mean I need to state the probability density function or something? And how do I show that $Y \sim N(0,1)$?
96591
Given that a random variable $X$ is a normally distributed variable with mean $\mu$ and variance $\sigma^2$. Why exactly is:  $$\mathbb{E}[|X|] = \mathbb{E}[X\mid X&gt;0]P(X&gt;0) - \mathbb{E}[X\mid X&lt;0]P(X&lt;0)$$  Because we have that:  $$\mathbb{E}[X\mid X&gt;0] = \frac{\int_0^\infty xf_X(x)\,dx}{P(X&gt;0)}$$  Therefore my definition immediately above must be incorrect?  Because if it is not we have that:  $$\mathbb{E}[|X|] = \int_0^\infty xf_X(x)\,dx - \int_{-\infty}^0 xf_X(x)\,dx$$  and where does the minus there come from anyway, and why?
73933
I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the random sample, after each random variable of the random sample has taken on a specific value?   $$(1) \ S=f(X_1,X_2...X_n)$$  $$(2) \ s=f(x_1,x_2...x_n)$$  I haven't been able to get any clarification for this and I've seen the term statistic describe both situations
108387
I read at  that      A random variable is a variable whose value depends on the outcome of a probabilistic experiment. Its value is a priori unknown, but it becomes known once the outcome of the experiment is realized.   That sounds to me like the definition of an observable in quantum mechanics modeled by hermitian operators. In addition it seems to me what statisticians call  of a random value is what physicists call eigenvalue of a hermitian operator. The set of realizations of a random variable would then be the  (set of eigenvalues) of an operator. So could I tell a statistician that a "random variable" is in fact an operator?
3091
This is a follow-on from my  about samples from a distribution.  Suppose $X_1 \ldots X_{n-1}, X_n$ are random variables all following some fixed distribution $D$. How do I prove that $P(X_n &gt; X_1 \wedge \ldots \wedge X_n &gt; X_{n-1}) = \frac{1}{n}$ (this is true intuitively and experimentally).
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Quantum Mechanics is very successful in determining the overall statistical distribution of many measurements of the same process.  On the other hand, it is completely clueless in determining the outcome of a single measurement. It can only describe it as having a "random" outcome within the predicted distribution.  Where does this randomness come from? Has physics "given up" on the existence of microscopic physical laws by saying that single measurements are not bound to a physical law?  As a side note: repeating the same measurement over and over with the same apparatus makes the successive measurements non-independent, statistically speaking. There could be a hidden "stateful" mechanism influencing the results. Has any study of fundamental QM features been performed taking this into account? What was the outcome?    Edit: since 2 out of 3 questions seem to me not to answer my original question, maybe a clarification on the question itself will improve the quality of the page :-)  The question is about why single measurements have the values they have. Out of the, say, 1000 measure that make a successful QM experiment, why do the single measurements happen in that particular order? Why does the wave function collapse to a specific eigenvalue and not another? It's undeniable that this collapse (or projection) happens. Is this random? What is the source of this randomness?  In other words: what is the mechanism of choice?    Edit 2: More in particular you can refer to chapter 29 of "The road to reality" by Penrose, and with special interest page 809 where the Everett interpretation is discussed - including why it is, if not wrong, quite incomplete.
58425
I'm having trouble making sense out of the formula $$(\hat{p}_1-\hat{p}_2) \pm 1.96 \sqrt{\frac{\hat{p}_1 (1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2 (1-\hat{p}_2)}{n_2}}$$  Is the idea that the sampling distribution of the difference between proportions of group $1$ and proportions of group $2$ of sample size $n$ is normally distributed regardless of population distribution shape, the same way it is true of the sampling distribution of the sample means, for example?  And in this case, what does that mean exactly? Say we have a sample size of $10$; are we talking about every possible combination of differences between every possible combination of samples of size $10$ for both groups and saying that that "function" is approximately normally distributed? What does it mean to subtract one sampling distribution from another?   And then, where does the square root come from? Is it just the SE of the distribution resulting from subtracting the sampling distribution of sample proportions of group 2 from the sampling distribution of group 1?
113644
It is understood in mathematical statistics that a sample (as in ) may very well contain repeatedly the same item/subject.  In practice though, it would never occur to someone conducting an experiment (e.g. in human sciences studies) to take into account the same subject twice. Samples are made    by taking the first people who agree to take the tests and look random enough (e.g. using quota sampling). by taking the only $12$ people living in your country who fit the requirements of the study (e.g. when you're conducting research on a rare disease) (and yes, $12$ is ridiculously small but that's still what is being done when no more subjects are available).      In what situations can this habit turn out to be really bad   - induce a huge bias on the results?   What seems clear to me is that if the total population is large enough, then it shouldn't matter too much, since samples with repeats are quite unlikely to occur and don't weigh much in the sampling distribution. When the total population is small though, it feels like we're more likely to miss something.
81879
As statisticians, we come across many distributions under the banners "discrete" or "continuous", and "univariate" or "multivariate". But can anyone provide a good reason behind the existence and motivation for so many distributions? How do we get them? And what can a layman understand from it?  What is the logic behind the existence of distributions?
111367
I know that "variable" means "values which vary." In a simple linear regression model :  $$Y=\beta_0+\beta_1X+\epsilon$$  $X$ is variable that is the values of $X$ vary. Why is $X$ not a random variable? What is the difference between a variable and a random variable?
1061
How can I prove that a random variable taking values in $[0,1]$ has variance no larger than $\frac{1}{4}$? If it matters, discrete and continuous proofs are both welcome.
74916
In a problem that I'm solving I find that:  "Let data (yi,xi) be sampled randomly from a two-dimensional distribution such that y|x is N(ɑ,x^2σ^2)".  Are y and x i.i.d? maybe just identically distributed but not independent?  Finally, if my model is: y = ɑ - u (where u is the error term and ɑ is a constant), and I also know that the distribution of x does not depend on either a or σ (I don't know it this assumption is needed for answer at the following question), is u independent from x?  My question is due to the fact that I thought that if the data are sampled randomly this mean that the r.v. are independent, but how is possible that the conditional distribution (his variance) depend then on x?
56775
In my probability class the terms "sums of random variables" is constantly used. However, I'm stuck on what exactly that means?  Are we talking about the sum of a bunch of realizations from a random variable? If so, doesn't that add up to a single number? How does a sum of random variable realizations lead us to a distribution, or a cdf / pdf / function of any kind? And if it isn't random variable realizations, then what exactly is being added?
316
In the textbook "Causal Inference for Statistics" by Rubin and Imbens, the following argument is made on pg. 39:     "In part of this text we view our sample of size N as a random sample   from an infinite super-population. In that case we employ slightly   different formulations of the restric- tions on the assignment   mechanism. Sampling from the super-population generates a joint   sampling distribution on the quadruple of unit-level variables (Yi(0),   Yi(1), Wi, Xi), i = 1, . . . , N. More explicitly, we assume the   (Yi(0), Yi(1), Wi, Xi) are independently and identically distributed   draws from a distribution indexed by a global parameter."   I am wondering why the bolded portion is needed to be assumed. In the context of observational studies what happens if 1) we have dependence, and 2) if the tuple is not jointly identically distributed.   Is this a general assumption that can be relaxed? Where do we really need it? Specifically:  1) What happens if we have dependence?  2) What happens if they are NOT identically distributed?
31013
Let $\mathcal{G} = \{A_1, \ldots, A_n\}$ be a partition of a set $\Omega$, $\mathcal{F} = \sigma(\mathcal{G})$. Prove that $X : \Omega\to\mathbb{R}$ is a random variable if and only if it is constant on each of the partition elements $A_j$.  I have attempted both directions but faced some difficulties to continue writing the proof. I have also consulted some sources that have a not so clear proof.  Here is the proof: We know that $$\sigma(\mathcal{G}) = \left\{\sum_{i\in I}A_i : I \subset \{1,\ldots,n\}\right\}.$$ ($\Leftarrow$) Let $B\in \mathcal{B}(\mathbb{R})$. In this case $X(\omega) = a_i$, $\forall\omega\in A_i$ for some constants $a_i$, (then I am not sure how to proceed to show that it is a random variable), the proof says that $X = \sum_{i\in I}^n a_i \mathcal{{1_A}_i}$ and $\{X \in B\} = \sum_{i : a_i\in \mathcal{B}}A_i\in \mathcal{F}$, so $X$ is a random variable.  My question is why are we concluding or defining $X = \sum_{i\in I}^n a_i \mathcal{{1_A}_i}$ (what is the purpose), in other words I don't understand it's flow of reasoning.  ($\Rightarrow$) We use proof by contradiction: If $X$ is not constant on $A_i$, then $\exists i_0$ such that for some $\omega_1, \omega_2 \in A_{i_0}$ one has $a:= X(\omega_1) \not= X(\omega_2)$. Then I am not sure how to proceed from here.  Many thanks in advance.