Activity for JRN‭

Type	On...	Excerpt	Status	Date
Question	—	Notation for nested exponents An expression such as $a^{b^c}$ is usually interpreted as $a^{(b^c)}$ and not as ${(a^b)}^c$. (See, for example, the Wikipedia entry for double exponential function.) Is there a reputable source that states how an expression such as $a^{b^{c^{\cdots^n}}}$ is to be interpreted? (more)	—	almost 2 years ago
Answer	—	A: Intuitively, why can $a, b$ cycle in ${\color{red}{b}} = \frac c{\color{red}{a}} \iff {\color{red}{a}} = \frac c{\color{red}{b}}$? "what's the intuition why a,b can swap places, whilst c remains in the numerator?" It's called the commutative property of multiplication. If $ab=c$ leads to $b=c/a$, then $ba=c$ leads to $a=c/b$. (more)	—	over 2 years ago
Answer	—	A: Is $r \times \frac{d}{dt} mv=\frac{d}{dt} (r \times mv)$ In equations 1.9 and 1.10, the quantities $\mathbf{r}$, $\mathbf{F}$, $\mathbf{N}$, and $\mathbf{v}$ are vectors, and the symbol $\times$ denotes a vector cross product operation. The product rule of differential calculus also applies to the vector cross product. That is, $\frac{\mathrm{d}}{\ma... (more)	—	over 2 years ago
Answer	—	A: What is a dummy variable (in an integral)? When integrating, a dummy variable is one that "disappears completely in the final result." For example, in the expression $\int x\ \mathrm{d}x$, $x$ is not a dummy variable because the expression is equivalent to $\frac{x^2}{2}+c$, that is, the $x$ hasn't "disappeared." If we had replaced $x$ wi... (more)	—	almost 3 years ago
Answer	—	A: Intuitively, why does A, B independent $\iff$ A, $B^C$ independent $\iff A^C, B^C$ independent? If events $A$ and $B$ are independent, then the probability that event $A$ happens is not affected by whether $B$ happens. If it isn't affected by whether $B$ happens, then it isn't affected by whether $B$ doesn't happen. (The rest of your statements can be shown to be true with the same reasoning.... (more)	—	almost 3 years ago
Answer	—	A: Why aren't the "21 possibilities here" NOT equally likely? If the $36$ ordered pairs are equally likely, then the probability of getting a cake cone with chocolate in the afternoon and a waffle cone with vanilla in the evening is $P($cakeC,waffleV$)=1/36$, and the probability of getting a waffle cone with vanilla in the afternoon and a cake cone with chocola... (more)	—	almost 3 years ago
Answer	—	A: "A occurred" vs. "something must happen" $A$ is a specific event, specified beforehand. Let's have an example. Let a random experiment be rolling a six-sided die and looking at what face lands up. The sample space is $S=\\{1,2,3,4,5,6\\}$. What is event $A$? Let's specify it before the experiment is done. Let $A$ be the event that... (more)	—	almost 3 years ago
Answer	—	A: How can Abraham Wald's approach lead you to ignore crucial features of a problem? If we add 1000 g of ethanol to 18 g of ethanol, the result has a weight of 1018 g. If we add 1000 mL of water to 18 mL of water, the result has a volume of 1018 mL. A person who focuses on the abstract (and not the concrete) would only look at 1000+18=1018 (the "struts and nails"); the other de... (more)	—	almost 3 years ago
Answer	—	A: Why are you permitted to define $1 − 1 + 1 − 1 + . . .$? By definition, a positive real number is a real number greater than zero. That statement cannot be proved to be right; it cannot be proved to be wrong. We either reject it and use a different definition, or we accept it and move on. Consider the statement "$1-1+1-1+... = 0$." Here is a "proof... (more)	—	almost 3 years ago
Answer	—	A: Why rational to be indifferent between two urns, when urn A has 50-50 red and white balls, but you don't know urn B's ratio? Say you have a coin and, if flipped, will land either heads or tails. What is the probability that it lands, say, heads? The "real" answer is that the probability is unknown. The information was not given at the start. We cannot proceed further then. But if we insist on moving on, we have to hav... (more)	—	almost 3 years ago