The Econonometrics book by Hansen mentions this property of a multivariate normal distribution: If $X∼N(\mu,\Sigma)$ then $AX+b∼N(Aμ+b,AΣA^\prime)$. It goes to say that a simple implication of this that each individual $X_j$ in $X$ is univariate normal. Could someone please explain how is this implied from the linearity property? Also is the converse true? If each $X_j$ in a $k$-vector random variable $X$ is normally distributed, is $X$ normally distributed?
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?
I'm planning a trip to recover a motorcycle from a distant city. The ideal trip would be flying to that location and coming back riding the bike, but that would require me to transport my riding gear with me in the plane: Leathers, boots, gloves, back protector and crash helmet.  All this is bulky and expensive stuff, and a crash helmet may be damaged if dropped or hit hard.   I don't expect any airline to allow me a crash helmet as hand luggage; and wearing it all the time is out of the question :-)   To complicate things, I would be coming back riding the bike, so any hard suitcase would have to be left behind.   How do people normally transport that kind of stuff? LWC Navigation: "this[f.NavigationMixin.Navigate] is not a function" When attempting to use the Lightning navigation service in my component's base class (.js), I get the following error when functionally testing in the UI:  this[f.NavigationMixin.Navigate] is not a function  I am calling the NavigationMixin function provided by the lightning/navigation module. I have imported the module: import { NavigationMixin } from 'lightning/navigation';  And am calling the function as below: this[NavigationMixin.Navigate]({     type: 'standard__recordPage',     attributes: {         recordId: this.org.Id,         objectApiName: 'Account',         actionName: 'view'     } }); My USB drive used to be originally 8GB when I bought it.  I'm trying to reformatted in Windows 7 by right clicking on the drive and selecting Format.... But the capacity only shows 250MB.  Is there something I can do to get the original size back? Maybe it got partitioned in a weird way?   The  flash drive is a SanDisk Cruzer Micro 8GB. I got it from Wal-Mart but this is the same drive. Continuum between addition, multiplication and exponentiation? I noticed this  which attempts to find the shades of grey between a linear and log scale where results are between zero and one.  However, I was looking for the more general case where we find the continuum between the operators themselves - add, multiply, and exponentiation or even higher.  The function could be a hyperoperation$(n,x,y)$, where $x$ and $y$ are the numbers to operate on, and $n$ is allowed to be a non-integer number for the type of operator. So $n=1$ would be 'addition', $n=2$ would be multiplication, and $n=3$ would be exponentiation. But one would also have $n=2.5$ to be the shade of grey BETWEEN multiplication and exponentiation, or $n=3.5$ or $4$ for beyond exponentiation and tetration).  I want the formula to work with real numbers, not just integers. Any thoughts?  These related posts may also be of interest: Copy files and directory tree for filesize in specified range I would like to make a small script copying files from different directories example /pp/01 /pp/02/ etc... to another destination with the same directory design. But only those files within the directory with a size larger than 88MB but smaller than 93MB...   I would use a for loop and the find function. Are there more efficient ways? And how can I implement the find function exactly. Superscript outside math mode What is the easiest way to superscript text outside of math mode?  For example, let's say I want to write the $n^{th}$ element, but without the math mode's automatic italicization of the th.  And what if I still want the n to be in math mode, but the th outside? How do I make Endermen spawn more frequently? I'm playing vanilla Minecraft, and am getting ready to head to the end.  I need Eyes of Ender (and hence Ender Pearls), and in my main hunting grounds (a large desert area) as little as one Enderman may spawn per night.  Is there any way to "promote" Enderman spawning so I can get pearls more quickly? Is there a built-in checksum utility on Windows 7? Is there a built-in checksum/hash utility on Windows 7? When can we use 'much' in affirmative clauses? One can contract I have to I've when have is a helping verb, e.g.     I've got an octopus in my pants.   Is contracting the main verb technically incorrect or merely antiquated? My father loves to say,     I've a month-old smoked shoulder I can cook for dinner,   and he sounds like a crazy old man when he does. I would like to know if I can shut him down by informing him that he is incorrect. :) During the latest gaming session our level 3 Rogue was Two-Weapon fighting with a Rapier in the main-hand and a scimitar in the off-hand.  At the completion of our gaming session I contested the application of the Two-Weapon fighting rules as it seemed unbalanced.  Once I got home I reviewed the PHB and various RPGStack and Reddit but couldn't find a concrete answer.  My interpretation of the PHB Reference rules below is that although any class/character can technically wield a weapon in their off-hand, only those who have been granted a Bonus Action for which no explicit bonus action have been defined would be allowed to actually attack.  So in the case of a Rogue the Bonus Actions granted under Cunning Action allow for specific actions (none of which are attack).  The alternate interpretation would be the rules under Two-Weapon Fighting, perhaps it should be written      you receive a bonus action to attack with a different light melee weapon that you’re holding in the other hand.   This would, in my opinion, explicitly grant the Bonus Action to any character/class who is Two-Weapon Fighting.  After thorough review does it boil down to application of the rules-as-written and determining a house-rule to apply in our campaign?    References  Page 189 of PHB : Bonus Actions  Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action. The Cunning Action feature, for example, allows a rogue to take a bonus action.      You can take a bonus action only when a special ability, spell, or other feature of the game states that you can do something as a bonus   action. You otherwise don’t have a bonus action to take.   You can take only one bonus action on your turn, so you must choose which bonus action to use when you have more than one available. You choose when to take a bonus action during your turn, unless the bonus action’s timing is specified, and anything that deprives you of your ability to take actions also prevents you from taking a bonus action.  Page 96 of PHB  A Rogue receives cunning action at level 2, You can take a bonus action on each of your turns in combat.      This action can be used only to take the Dash, Disengage, or Hide action.   Page 195 of PHB  Two-Weapon Fighting (Combat Rules)  When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative. If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.  Page 165 of PHB  Dual Wielder  (Feat)  You master fighting with two weapons, gaining the following benefits:   You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand. You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren't light. You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one. What is the daily reputation cap and how can I hit it? I've found some questions and those two badges "Epic" and "Legendary" regarding the "daily reputation cap". What is it and how can I hit it? How do you prove the second law of thermodynamics from statistical mechanics? Is the 5-foot range of the Help action my range to the ally, the enemy, or both? Triangular Factorials I came across a statement online and have been looking for a proof :  It states that 1, 6 and 120 are the only numbers which are both triangular and factorials.  Is there any way I can prove this? This claim looks too 'big' and I've tried to prove it but  I couldn't. Can anyone help me to prove this? Reference to a figure uses the section's number Do immature fruits perform photosynthesis? Most immature fruits are green: peppers, pine cones, plums, lots of them. I want to know if the green is from chlorophyll in the cells. Do the fruit cells perform photosynthesis? When you cover a green stem or leaf, it will turn pale and stretch. That is because the stems have little need for chlorophyll in the dark, which is why they are pale. They stretch because the auxins in the stems are not destroyed by the photons, and the stems stretch out and topple over. If a green fruit is covered, will it turn pale and stretch like that? Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$ I was solving a few problems from a textbook and I came across this one:     Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through   the point $P=(12,3)$   I could find the tangency points $$ but my solution was completely messy (I had four equations that I had to go back-and-forth with.  Is there an elegant/simple solution to that?  Tangency points:  $$P_1 = (0,3)$$ $$P_2 = \left( \frac{24}{5} , 3-\frac{24}{5} \right)$$ Let $a$ and $b$ be two integers such that $\left(a,b\right) = 1$. Prove that $\left(a+b, ab\right) = 1$.   $(a,b)=1$ means $a$ and $b$ have no prime factors in common  $ab$ is simply the product of factors of $a$ and factors of $b$.  Let's say $k\mid a+b$ where $k$ is some factor of $a$.  Then $ka=a+b$ and $ka-a=b$ and $a(k-l)=b$.  So $a(k-l)=b, \  a\mid a(k-1)$ [$a$ divides the left hand side] therefore $a\mid b$ [the right hand side].  But $(a,b)=1$ so $a$ cannot divide $b$.  We have a similar argument for $b$.  So $a+b$ is not divisible by any factors of $ab$.  Therefore, $(a+b, ab)=1$.  Would this be correct? Am I missing anything? If I insert a flash drive and try to drag any files onto desktop - nothing happens.  Or from any other folder.  Drag 'n copy only works if you open desktop in nautilus ("show desktop in files").  I already ran the command:  gsettings set org.gnome.nautilus.preferences use-experimental-views false   It didn't help. How to fix this?