Everywhere it is done for bivariate or given a hints. But Not in details for trivariate.
121152
Let $Y$ ~ $MVN_3(\mu, \Sigma)$ where $\mu = (5,6,7)$ and $\Sigma = \begin{bmatrix}2 &amp; 0 &amp; 1\\0 &amp; 3 &amp; 2\\1&amp;2&amp;4\end{bmatrix}$ Find (a) The marginal distribution of $Y_1$ (b) The joint distribution of $Y_3$ and $Y_1$ (c) The conditional distribution of $Y_3$ given $Y_1$. Now, my experience with probability prior to this class is limited to Sheldon Ross's intro book and Wackerley, Mendenhall, and Schaeffer, which I'm currently using for my Into Math Stat class. I've never before done conditional, joint, or marginal densities with matrices, nor have I covered them for variables that are dependent.... So please forgive the embarassing attempts I have made to solve this. It also does not help that the class this problem is for does not have a textbook, just some notes frantically scrambled onto a whiteboard twice a week while our prof reminds us that he hates teaching and would rather be researching... My attempts (a) I can write $Y_1 = AY + b$, where $b=(0,0,0)^T$ and $A = (1,0,0)^T$. Thus: $Y_1$~$MVN(\cdot,\cdot)$ because it is a linear transformation of a multivariate normal. Thus: $E[Y_1]=E[AY+b]=AE[Y]+b=(1,0,0)^T(5,6,7)^T=5$ $Cov(AY) = ACov(Y)A^T = A\begin{bmatrix}2 &amp; 0 &amp; 1\\0 &amp; 3 &amp; 2\\1&amp;2&amp;4\end{bmatrix}A^T$ This becomes: $(2,0,1)\begin{bmatrix}1\\0\\0\end{bmatrix}=2$ However, knowing that $Y_1$ and $Y_3$ are not independent makes me question this result...    (b) The joint distribution of $Y_1$ and $Y_3$ I basically did the same thing as above but with $A = \begin{bmatrix}1 &amp; 0 &amp; 0\\0&amp;0&amp;1\end{bmatrix}$. This yielded $W = (Y_1, Y_3)^T$ ~ $MVN(\mu=(5,7)^T, \Sigma = \begin{bmatrix}2 &amp; 1\\1&amp;4\end{bmatrix})$ (c) The conditional distribution of $Y_3$ given $Y_1$ This part I'm honestly at a loss for...unless I'm just supposed to take the integral of the density function for $(Y_1, Y_3)$ divided by the density function for $(Y_1)$, i.e., conditional = $\int f_{Y_1,Y_3}/f_{Y_1} dy_1$. If I am supposed to do this, how do I get the actual joint density function and perform the integration and division? The only formulas I've been provided for multivariate normal distributions involve matrices, and I have never seen matrix integration. I apologize if my comprehension is lacking, my classmates and I have been thrown into this material with no assistance from our professor.
42988
There have been a few cases in which the  was not awarded.   There have also been other cases in which it was awarded incorrectly, on old questions.   What gives?
3361
Consider a statistical problem where you have a response variable that you want to describe conditional on an explanatory variable and a nested variable, where the nested variable only arises as a meaningful variable for particular values of the explanatory variable.  In cases where the explanatory variable does not admit a meaningful nested variable, the latter is usually coded either as NA in the data set, or if it is coded with a value, that value is merely a placeholder that does not have any meaningful interpretation.  This situation tends to arise whenever you have an explanatory variable indicating the existence of a thing, and one or more nested variables describing the characteristics of that thing.  Some examples of this kind of situation in statistical problems are the following:   The explanatory variable is an indicator of whether a survey participant is married, and the nested variable is some characteristic of the spouse (e.g., education, age, etc.); The explanatory variable is an indicator of the presence of an item in a space, and the nested variable is a measure of some characteristic of the item (e.g., size, distance, etc.); The explanatory variable is an indicator of the occurrence of an event and the nested variable is a description of some characteristic of the event (e.g., duration, magnitude, etc.).   In these kinds of situations, we often want to build a regression-type model (in the broad sense that includes GLMs, GLMMs, etc.) describing the relationship between the response variable and the other variables.  It is not obvious how to deal with the nested variable in this type of model.  Question: How do we deal with the nested variable in this type of model?    Note: This question is designed to give a generalised answer to a recurring question on  regarding nested variables in regression (see e.g., , ,  and ).  This question is designed to give a generalised context-independent example of this problem.
99500
Sometimes, in a math paper, we read something like "Although a priori this statement could be true, it is in fact false, as proven by this theorem". But aren't all mathematical statements a priori? So why include that misleading bit of language in the beginning?
18308
I wonder if we can we say $x\in \{\{\{x\}\}\}$?    In one viewpoint the only element of $\{\{\{x\}\}\}$ is $\{\{x\}\}$. In the other viewpoint $x$ is in $\{\{\{x\}\}\}$, for example all people in Madrid are in Spain.
39340
It seems that a number of the statistical packages that I use wrap these two concepts together. However, I'm wondering if there are different assumptions or data 'formalities' that must be true to use one over the other. A real example would be incredibly useful.
76095
I came across the following problem in a book:  Give a combinatorial proof of  $$ {n \choose 0} + {n \choose 2} + {n \choose 4} + \, \, ... \, = {n \choose 1} + {n \choose 3} + {n \choose 5} + \, \, ...$$ using the "weirdo" method (i.e., where one of the elements is chosen as special and included-excluded -- I'm sure you get the idea).  After days of repeated effort, the proof has failed to strike me. Because every time one of the elements is excluded, the term would be ${n-1 \choose k}$ and not $ {n \choose k}$, which is not the case in either of the sides of the equation.
32266
What I am thinking was I need two formulas, AX.p(X) => something AX.p(X) => ~ something   I guess something maybe is the p(x) and the other is ~p(x) since we was given EX.~p(x)..But actually it can't work for me. And I guess maybe is EX.~p(x) and ~EX.~p(x)..But if I am doing that then the it will go to deadend..so I give up..;(  Wish someone can help me.   For note about EE: 1: EX.p(X) 2: AX.(p(X) => something) 3: something EE1,2 But something cann't have the free variable from p..in this sample, then soemthing cannot including X..so if something is p(X) it cannot use EE, but if p(Y) then can use EE.
19743
I know that it is defined such that there is no more than one y-value output for any given x-value input, but I"m wondering WHY it is defined that way? Why can't we apply everything we know about functions to equations that have more than one y-value output for any given x-value input?   In short, what is the benefit or reason to define a function this way?  At the same time, a little historical context would also be appreciated (i.e., when the concept originated and why).  Thanks,  Moshe
47075
Or, in other words, why aren't all functions surjective? Isn't any subset of the codomain which isn't part of the image rather arbitrary?
28197
Somebody asked me this question in a job interview and I replied that their joint distribution is always Gaussian. I thought that I can always write a bivariate Gaussian with their means and variance and covariances. I am wondering if there can be a case for which the joint probability of two Gaussians is not Gaussian?
33015
I have been reading many articles on LASSO regression, and everyone claims that LASSO address multicollinearity showing contour plots of cost function touching the corner of the diamond(x1+x2). In the first place, I didn't understand how does this phenomenon translate to addressing multicollinearity when there are many groups of correlated variables in the data.  Verbatim from one of the research paper: "• If there are grouped variables (highly correlated between each other) LASSO tends to select one variable from each group ignoring the others".
120459
Does the following qualify as a univariate regression?  $$y=b_0+b_1x+b_2x^2+\epsilon$$  I fully comprehend the implications of adding regressors and need no background information - a "yes" or "no" will do :-)
56973
For example, this question might be offtopic. But where it is not?
86454
I am working on a paper, and have a problem regarding ellipsoid and Trivariate Normal Distribution. Suprisingly I can't find much in literature but I found your in one of your answers:     Because this construction has nothing to do with "confidence" per se,   the objective is to establish some convention for describing the shape   and relative size of the points. Using 1.96 sort of works (for three   variables): it contains about 72% of the probability of the trivariate   normal distribution. But as the number of variables increases this   method produces ellipses that are far too small. For instance, with 10   variables it will contain only 4.6% of the probability; using 4.28   instead of 1.96 in this case will contain 95% of the probability.   How did you get this number 72%? Or do you have some literature to recommend to me in which I can find this. I would appreciate it very much!
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?
31594
I guess the answer is NO, for polynomials.  I know that there are more than one series expansion for every function.  But I am talking about power series here.  All Ideas are appreciated