Determining the distribution of fitted values and residuals For exam prep, we're supposed to answer this question (which we are supposed to be able to find the answer to in the course literature)     For the model specified in a) and proper normality assumptions on ε, obtain the distribution of ŷ and e = y − ŷ   where the model referred to in a) is the standard form of the multiple regression model  \begin{equation} \mathbf y = \mathbf X \pmb \beta + \pmb \epsilon \end{equation}  with all proper normality assumptions fulfilled. I've been trying to find the answer in the chapter covering multiple regression in Introduction to Linear Regression Analysis (which is our course literature) and online, but I can't seem to find the answer.  Could you point me in the right direction?
Distribution of linear combination of OLS regression coefficients I have a simple linear OLS regression $Y_i = \alpha+ \beta_1 X_{1i} + \beta_2 X_{2i} + e_i$ where $e_i \sim N(0,\sigma)$. I have estimated the regression from the data and obtained estimates for my coefficients as well as the corresponding covariance matrix. Assume my dataset is large (>500)  Now I would like to construct the 95% confidence interval for $Z = a_1 \beta_1 + a_2 \beta_2$. My first thought was that I could assume that Z follows a normal distribution because my regression coefficients are approximately normal and a linear combination of normal variables is also a normal variable. I do find some sources where they treat $Z$ as a normal random variable.  But now I am in doubt because I found that a linear combination of normal random variables is only normal if they are independent. So is it still save to assume $Z$ is random?
I am wondering what sudo apt-get update does?  What does it update? why Booleans slow my computer more than expected Natural Sort Order in C# Computing $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$ What is the result of  $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$ . I tried to do couple of algebraic manipulations, but I didn't reach to any conclusion.  Thanks a lot. Integral representation $B_T^3$ I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me? How can I search for a keyword with special characters in Google Search? How can I read input from the console using the Scanner class in Java? How can I easily share links or text between my Android phone and my laptop? There are $2n+1$ people. For each $n$ people there is somebody who is friend with each of them. Prove there is a "know-them-all" person. I have a following problem.     There are $2n+1$ people in the room. For any group of $n$ people there is a person (different from them) who is friends with each person in this group. Prove that there is a person, who knows all $2n$ other people.   One can easily see, that $\min_v \deg v = n$. From this we can get a lower bound for number of edges $N_e$: $$ N_e = \frac{1}{2}\sum_v \deg v \geq \frac{(2n+1) \cdot n}{2} $$ I do not know where to move from this point (and do not think, that this is the right direction) and pretty stuck right now. Can you give a hint? Tourist to F1 change of status within 60 days I have a PhD place at a US university starting Aug 21 but want to visit a different university for casual research for 6 weeks from July 1. This is longer than the 30 day grace period for the F1. Can I arrive on the visa waiver program or tourist visa in July and then just stay on for the PhD? I guess I might need to change status. Will this work?  Please don’t suggest exiting and reentering, that’s too expensive.  (British masters student in UK) In the last 30 years, human technology has developed massively.  Every year a new smartphone is released with a faster processor, more memory, better features.  Why hasn't the same level of progress happened in the Star Wars universe?  In The Phantom Menace, we see a number of droids - R2 units, C-3PO, Pit Droids, etc.  In Return of the Jedi - the droids we see pretty much have the same capabilities.  I would have thought that technology would still be progressing in the Star Wars universe - but it appears to have stopped.  During wars on Earth, our technology capabilities grow - so it can't just be the effect of a massive war which is causing the stagnation of development.  So, what gives? And when I tried the following commands /etc/hostnames and /etc/hosts, it says:  bash: permission denied. How to set a domain to ip address? You can reach this IP address  (if my computer is online). I want to assign a domain to it so I wonder how is that possible when I have no DNS? I do own my domain which is ".com". My IP is static.  Thanks What could we call a person with deep knowledge in various fields? another balls and bins question I've seen many variations of this problem but I can't find a good, thorough explanation on how to solve it. I'm not just looking for a solution, but a step-by-step explanation on how to derive the solution.  So the problem at hand is:   You have m balls and n bins. Consider throwing each ball into a bin uniformly and at random.    What is the expected number of bins that are empty, in terms of m and n? What is the expected number of bins that contain exactly 1 ball, in terms of m and n?   How would I approach solving this problem?  Thanks! Does a multiple-entry Schengen visa allow visiting other Schengen countries later, without going via the issuing country? Antecedent Precedence? Resolving ambiguous overload on function pointer and std::function for a lambda using + Is there any way I can use  partitions on Windows&nbsp;7?  I have tried to install , but it doesn't seems to run on Windows&nbsp;7. I have a .net client application which is connected to a remote database. Is it safe to keep a single connection open for the lifetime of the client (hours)? Does the answer hold if I have multiple (10 or 100) clients running?