https://math.codidact.com/categories/41/feedNew Posts - Q&A - MathematicsMathematics - Codidact2021-10-22T14:26:48Zhttps://math.codidact.com/posts/284651What does the distribution of a strictly increasing random walk look like if you stop at `N` steps?SherlockSagehttps://math.codidact.com/users/537952021-10-22T13:48:49Z2021-10-22T14:26:48Z<p>I have a puzzle inspired by the game Idle Brick Breaker (<a href="https://play.google.com/store/apps/details?id=com.TechTreeGames.IdleBrickBreaker&hl=en&gl=US">Google Play</a> and <a hre...https://math.codidact.com/posts/284557Does this construction always give a topological vector space?celtschkhttps://math.codidact.com/users/80562021-10-10T08:42:10Z2021-10-10T19:54:07Z<p>Consider an algebraic vector space $V$ over $\mathbb R$ or $\mathbb C$.</p>
<p>Now for each possible basis $B_k = \{e_i^{(k)}|i\in I\}$ of $V$, one can define an inner product $\langle\cdot\ver...https://math.codidact.com/posts/284550transpose matrix and general matrix is completely messed upIstiakhttps://math.codidact.com/users/539222021-10-09T16:01:33Z2021-10-10T18:17:17Z<p>I was studying Matrix. I never attend any lecture. So I don't have much more idea of Matrix. I had learned multiplication, addition, subtraction and inverse matrix last year from some YT tutoria...https://math.codidact.com/posts/284561What does variable in sup represent for matrix?Istiakhttps://math.codidact.com/users/539222021-10-10T15:08:24Z2021-10-10T15:08:24Z<p>I had seen people were representing matrix 2 way.</p>
<ol>
<li>$$\sum_{j=1}^n a_{ij}$$</li>
</ol>
<p>It is representing a column matrix (vector actually) if we assume $i=1$.</p>
<p>$$\begin...https://math.codidact.com/posts/284408What it ≡ called?Istiakhttps://math.codidact.com/users/542122021-09-30T16:23:00Z2021-09-30T16:38:32Z<p>I saw the text (≡) few times. When I had searched in internet I got that..</p>
<blockquote>
<p>In mathematics, the triple bar is sometimes used as a symbol of identity or an equivalence relati...https://math.codidact.com/posts/284279$\int dx dy dz d p_x dp_y dp_z$ Does it have any physical meaning?Istiakhttps://math.codidact.com/users/539222021-09-21T16:02:49Z2021-09-22T21:59:36Z<p>I was reading a Physics book. Then I saw an equation which was looking like this :</p>
<p>$$\int dx dy dz d p_x dp_y dp_z$$</p>
<p>I was thinking it from just a Calculus book. I can see lots o...https://math.codidact.com/posts/284280Is Pythagorean theorem really valid in higher dimensional space?Istiakhttps://math.codidact.com/users/539222021-09-21T16:25:27Z2021-09-22T02:17:59Z<p>I saw that someone was writing Pythagorean theorem in 3 dimensional space. The equation was :</p>
<p>$$c=\sqrt{x^2+y^2+z^2}$$
If it's really correct than Pythagorean should work in higher dime...https://math.codidact.com/posts/282643If A & B are joint, can Arby recoup some of his loss only when $P_{Arby}(A \cup B) < P_{Arby}(A) + P_{Arby}(B)$?DNBhttps://math.codidact.com/users/536282021-07-11T06:07:51Z2021-09-18T08:18:31Z<ol>
<li>
<p>Please see the sentence alongside the red line below, but the authors didn't write this sentence for the first case ( $P_{Arby}(A \cup B) < P_{Arby}(A) + P_{Arby}(B)$). Thus if A ...https://math.codidact.com/posts/282966What's wrong with evaluating $n(n-1) \dots (n-[k-3])(n-[k-2])\color{red}{(n-[k-1])}$ at $k = 1$? DNBhttps://math.codidact.com/users/536282021-07-30T02:00:37Z2021-09-18T06:35:14Z<p>This snag arose out of <a href="https://math.codidact.com/posts/282606">this post</a>, and <a href="https://math.codidact.com/comments/thread/3555#comment-11900">these comments by r~~</a>. In th...https://math.codidact.com/posts/282645If Alice must've have classes on at least 2 days, why do you need the intersection of 3 $A_i^C$'s?DNBhttps://math.codidact.com/users/536282021-07-11T06:42:50Z2021-09-18T06:01:51Z<p>Can someone please rectify my MathJax? Please see the red phrase below.</p>
<ol>
<li>
<p>The question itself never touts or postulates outright that Alice "must have classes on at least 2 da...https://math.codidact.com/posts/283260How would you vaticinate to $-w_k$ from both sides of $w_{k + 1} = \dfrac{w_k - (1 - p)w_{k - 1}}{p}$?DNBhttps://math.codidact.com/users/536282021-08-10T06:25:17Z2021-09-16T16:22:40Z<p>This question appeared on my pop quiz last week. I got 0%. I achieved everything until the green equation, then I didn't know how to proceed. After reading this solution, I see that you must iso...https://math.codidact.com/posts/283872Getting backward of partial differentiation's chain ruleIstiakhttps://math.codidact.com/users/539222021-08-31T15:08:18Z2021-09-13T13:09:59Z<p>We know that Chain rule of partial derivatives is something just like this ($z$ is function of $x$ and $y$ variable and, $x$ and $y$ is function of $t$) :</p>
<p>$$\frac{dz}{dt}=\frac{\partial ...https://math.codidact.com/posts/283252Which vertical line signifies "putting the cutoff for a positive result at a very low level"?DNBhttps://math.codidact.com/users/536282021-08-09T23:44:39Z2021-09-12T05:14:47Z<p>The author, Karen Stewart <a href="https://www.linkedin.com/in/karen-steward-89036a24/">MA Natural Sciences (Univ. of Cambridge) PhD Veterinary Microbiology (Cambridge)</a>, refers to "a purple ...https://math.codidact.com/posts/284113What to do when there's lot of function of time? (For integration) Should I consider them as constant?Istiakhttps://math.codidact.com/users/539222021-09-10T09:03:12Z2021-09-10T18:11:41Z<p>Let I have a function which is looking like this</p>
<p>$$f=\int\frac{l \ \mathrm {dt}}{\dot{\theta}r\dot{r}}$$</p>
<p>Here $\dot{\theta}$, $r$, $\dot{r}$ and $l$ all of them are function of $...https://math.codidact.com/posts/283929Consider the second of these integrals (What's the meaning of second right here?)Istiakhttps://math.codidact.com/users/539222021-09-02T10:51:07Z2021-09-03T10:32:57Z<p>$$\frac{dJ}{d\alpha}=\int_{x_1}^{x_2}(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha}+\frac{\partial f}{\partial \dot{x}}\frac{\partial \dot{x}}{\partial \alpha})\mathrm dx$$
Co...https://math.codidact.com/posts/283945What're the orders for equation expressing?Mark Giraffehttps://math.codidact.com/users/541142021-09-02T23:59:30Z2021-09-03T04:28:09Z<p>When doing an equation in programming or real calculations, equation expression takes in a specific order. In a single equation, I use the PEMDAS/GEMDAS order, which goes:</p>
<ul>
<li>Perform...https://math.codidact.com/posts/283111Why's the true positive rate termed Sensitivity and true negative rate Specificity, not vice versa? DNBhttps://math.codidact.com/users/536282021-08-05T02:11:34Z2021-09-02T22:39:51Z<ol>
<li>
<p>To wit, what's "Sensitive" about True Positive Rates, and "Specific" about True Negative Rates?</p>
</li>
<li>
<p>Why weren't these <a href="https://en.wikipedia.org/wiki/Semantic...https://math.codidact.com/posts/282604Why aren't the "21 possibilities here" NOT equally likely? DNBhttps://math.codidact.com/users/536282021-07-09T03:59:27Z2021-09-02T14:15:09Z<p>Please see the last sentence below, that I highlighted in red.</p>
<hr>
<p>Example $1.4 .5$ (Ice cream cones). Suppose you are buying an ice cream cone. You can choose whether to have a cake c...https://math.codidact.com/posts/282603Without calculations, how can you visualize "that half the squares are white and half are black"?DNBhttps://math.codidact.com/users/536282021-07-09T03:50:39Z2021-09-02T14:14:35Z<p>Please see the 2nd para. below alongside my red highlighted words. I can't "[i]magine rotating the chessboard 90 degrees clockwise." I can't visualize how "all the positions that had a white squ...https://math.codidact.com/posts/282888How can I visualize the Law of Total Probability with extra conditioning?DNBhttps://math.codidact.com/users/536282021-07-28T00:06:49Z2021-09-02T14:14:13Z<p>How can I pictorialized this Theorem 2.4.3? As you can see below, I edited a picture by drawing E inside B.. Is my edit correct? Can my edit be improved?</p>
<blockquote>
<p><strong>Theorem 2....https://math.codidact.com/posts/281008If a repair on a vessel costs $2$ and fully repaired vessel is worth $1$, don't we need know worth of broken vessel to decide whether to repair?Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:37:11Z2021-09-02T14:12:49Z<p>I divided all prices by 100,000 to simplify the question. The book beneath doesn't stipulate the current value of the vessel. But don't we need to know it, to calculate if it's worth repairing? ...https://math.codidact.com/posts/282889How do you calculate $P(X = n|G), P(X = n|G^C)$ by the Law of Total Probability, with extra conditioning? DNBhttps://math.codidact.com/users/536282021-07-28T02:15:02Z2021-09-02T14:12:22Z<ol>
<li>
<p>Please see $P(U|G)$ and $P(U|G^C)$ below, beside my red line. Can you please expound these calculations?</p>
</li>
<li>
<p>How was $s$ computed in both equations?</p>
</li>
</o...https://math.codidact.com/posts/282614Why $\color{red}{k\dbinom{k}{1}} \neq$ "first choose the k team members and then choose one of time to be captain"?DNBhttps://math.codidact.com/users/536282021-07-09T06:38:52Z2021-09-02T14:11:53Z<p>Because you "first choose the k team members and then choose one of time to be captain", shouldn’t the RHS be $\color{red}{k\dbinom{k}{1}}$? The captain is chosen from the $k$ team members alre...https://math.codidact.com/posts/283887Does "both girls, ≥ 1 winter girl" = "both girls, ≥ 1 winter child" let you generalize the problem statement that postulated both children as girls? DNBhttps://math.codidact.com/users/536282021-09-01T08:40:20Z2021-09-01T08:40:58Z<p>What does "the fact that "both girls, at least one winter girl" is
the same event as both girls, at least one winter child"" imply about the problem statement below that posited merely girls? I...https://math.codidact.com/posts/283886How would you intuit, and soothsay to rewrite, "both girls, ≥ 1 winter girl" as "both girls, ≥ 1 winter child"?DNBhttps://math.codidact.com/users/536282021-09-01T08:35:52Z2021-09-01T08:37:39Z<ol>
<li>
<p>Please see the red underline. How can you intuit that "both girls, at least one winter girl" as "both girls, at least one winter child"? I ask this for my 15 y.o.</p>
</li>
<li>
<...https://math.codidact.com/posts/283388How do I prove Simpson's Paradox, scilicet $P(A|B) > P(A|B^C)$?DNBhttps://math.codidact.com/users/536282021-08-14T08:22:59Z2021-08-27T09:10:11Z<p>$\forall a,b,c,d > 0, a<b, c<d \implies$ <a href="https://math.stackexchange.com/q/3632752">$0 \le ab < cd$</a>.</p>
<ol>
<li>
<p>The blue, orange and purple inequalities below ar...https://math.codidact.com/posts/283696Find limits of integration in polar coordinatesMissMulanhttps://math.codidact.com/users/541072021-08-24T13:36:10Z2021-08-25T15:22:45Z<p>Suppose we have this graph and we want to find the area between the 2 curves:</p>
<p><img alt="hi" src="https://math.codidact.com/uploads/xum62GPERZqYiqMuiVmgLAq1">.</p>
<p>We can calculate th...https://math.codidact.com/posts/283385How does counting E twice explain the discrepancy between the third between C and E, third between E and G v. fifth between C and G?Chgg Clouhttps://math.codidact.com/users/535642021-08-14T03:55:21Z2021-08-25T12:09:04Z<p>I still don't grasp the "source of the discrepancy". "the E got counted twice when we went C,D,E and then E,F,G, but only got counted once when we went C,D,E,F,G." — So what? How does this expou...https://math.codidact.com/posts/283255After $n - 2$ unchosen doors are opened, how does the probability of the $n - 2$ unchosen doors "shift" or "transfer" to the lone unchosen door?DNBhttps://math.codidact.com/users/536282021-08-10T00:43:39Z2021-08-24T06:08:04Z<p>The second quotation below uses the verb "shift" to describe how Monty Hall's opening the 98 unchosen doors (revealing a goat each) ""shifts" [boldening mine] that 99/100 chance to door #100"? T...https://math.codidact.com/posts/283633Is $r \times \frac{d}{dt} mv=\frac{d}{dt} (r \times mv)$Istiakhttps://math.codidact.com/users/539222021-08-21T12:09:40Z2021-08-22T02:33:29Z<p>Is $$r \times \frac{d}{dt} mv=\frac{d}{dt} (r \times mv)$$</p>
<p>I was thinking that it is wrong. Cause, which is outside of differentiation how we can put that inside differentiation no matte...https://math.codidact.com/posts/283118Why would skyrocketing the numbers of doors help laypeople intuit the Monty Hall Problem? DNBhttps://math.codidact.com/users/536282021-08-05T04:27:25Z2021-08-20T23:17:49Z<p>Alas, it isn't clear to me that <strong>it becomes clear that the
probabilities are not 50-50 for the two unopened doors</strong>. Had I never seen this exercise or problem, even if there were ...https://math.codidact.com/posts/282942Besides guaranteeing 0 ≤ P(A ∩ B) ≤ 1, why can't Independence be defined as P(A ∩ B) = P(A) +,-, or ÷ P(B)? DNBhttps://math.codidact.com/users/536282021-07-29T08:02:00Z2021-08-20T23:10:11Z<p>I know that $0 \le P(A \cap B) \le 1$ will be violated if $P(A \cap B) = P(A) +,-,\text{or} ÷ P (B)$. I'm not asking about or challenging this reason.</p>
<p>But what are the other reasons agai...https://math.codidact.com/posts/283586What is "continuous" in Math?JohnDoeahttps://math.codidact.com/users/363632021-08-19T11:30:46Z2021-08-20T03:43:34Z<p>I understand that in Math, there is a common separation between <em>discrete</em> and <em>continuous</em>.</p>
<ul>
<li>I understand that in Math we could say that <em>discrete is anything (an...https://math.codidact.com/posts/283400If a matrix has lots of values in a column but, not in row than, what that actually called?Istiakhttps://math.codidact.com/users/539222021-08-14T16:40:29Z2021-08-18T15:31:29Z<p>$\begin{bmatrix}a & b & c & d\end{bmatrix}$</p>
<p>$\begin{bmatrix}a \\ b \\ c \\ d \end{bmatrix}$</p>
<p>Which one is $1$ dimensional Matrix? I think that first matrix is $1$ dime...https://math.codidact.com/posts/283513How to find constant equal to what in integration?Istiakhttps://math.codidact.com/users/539222021-08-17T08:27:11Z2021-08-18T15:14:59Z<p>$$x(t)=\int \dot{x}(t)\mathrm dt=vt+c$$</p>
<p>That's what I did. But, book says</p>
<p>$$x(t)=\int \dot{x}(t)\mathrm dt=x_0+v_0 t+ \frac{F_0}{2m}t^2$$</p>
<p>Seems like, $x_0 + \dfrac{a_0}{2...https://math.codidact.com/posts/283516Is $x=\int \int \ddot{x}\mathrm dx \mathrm dx$?Istiakhttps://math.codidact.com/users/539222021-08-17T08:47:58Z2021-08-17T12:18:18Z<p>I know that multivariable calculus used like this</p>
<p>$$\phi = \int \int x \mathrm dx \mathrm dy$$</p>
<p>But, I was thinking to use double integral for single variable (double integral res...https://math.codidact.com/posts/283518Do these major triangle lines have names?Peter Taylorhttps://math.codidact.com/users/363562021-08-17T11:12:59Z2021-08-17T11:12:59Z<p>Kimberling's <a href="https://faculty.evansville.edu/ck6/encyclopedia/ETC.html">Encyclopedia of Triangle Centers</a> lists, among other things, lines on which each centre is found, but usually l...https://math.codidact.com/posts/283445Prove that $\int \ddot{x}(t)\mathrm dt=v_0 + \frac{F_0}{m}t$Istiakhttps://math.codidact.com/users/539222021-08-16T10:38:03Z2021-08-16T13:13:41Z<p>$$\ddot{x}(t)=\frac{F_0}{m}$$
This is a second-order differential equation for x (t) as a function of t. (Second-order
because it involves derivatives of second order, but none of higher order...https://math.codidact.com/posts/283398Finding the smallest Mersenne-number multiple of an odd integerwizzwizz4https://math.codidact.com/users/530362021-08-14T16:20:46Z2021-08-16T07:35:16Z<h3>Problem</h3>
<p>Given an odd integer $a$, there exists some smallest $b$ such that $a\times b = 2^n-1$ (for some $n$). Find $b$ and $n$.</p>
<ul>
<li>What is an upper bound for this problem?...https://math.codidact.com/posts/283372How can I intuit $\dfrac{a - b}{c - d} \equiv \dfrac{{\color{red}{-}}(b - a)}{{\color{red}{-}}(d - c)} \equiv \dfrac{b - a}{d - c}$? Chgg Clouhttps://math.codidact.com/users/535642021-08-13T07:01:36Z2021-08-14T18:38:58Z<p>I'm not asking about algebra here which I can effortlessly effectuate. If helpful, let's <a href="https://betterexplained.com/articles/subtracting-negative-numbers/">intuit subtraction as facing...https://math.codidact.com/posts/283390Book suggestion for Tensor Calculus (Differential geometry) Istiakhttps://math.codidact.com/users/539222021-08-14T10:16:00Z2021-08-14T15:26:23Z<p>I have studied Multi-variable calculus. Now, I am thinking to study Tensor calculus. As it is an important topic in <a href="https://en.wikipedia.org/wiki/General_relativity">General Relativity<...https://math.codidact.com/posts/283295In $w_{k + 1} - w_k = (\frac{1 - p}{p})^{exponent}(w_1 - w_0)$, why isn't exponent $k + 1$?DNBhttps://math.codidact.com/users/536282021-08-11T08:23:25Z2021-08-14T13:43:51Z<p>Please see the $r^k$ underlined in red, which is $(\frac{1 - p}{p})^k$ as defined by the green underlines.</p>
<ol>
<li>
<p>How do you deduce that the exponent must be $k$? Why isn't the expo...https://math.codidact.com/posts/283387Why "only 1/10,000 men with wives they abuse subsequently murder them" ≠ P(A|G,M) & "50% of husbands who murder their wives abused them” ≠ P(G|A)?DNBhttps://math.codidact.com/users/536282021-08-14T07:33:58Z2021-08-14T07:33:58Z<p>All emboldings are mine. See my red side line — the solution identifies <strong>"only 1 in 10,000 men with wives
they abuse subsequently murder their wives"</strong> (in the problem statement) ...https://math.codidact.com/posts/283371How can I exploit symmetry to intuit P(A wins) + P(B wins) = 1, without performing algebra? DNBhttps://math.codidact.com/users/536282021-08-13T06:55:38Z2021-08-13T06:55:38Z<p>I'm not asking about algebra that I can execute. How can I intuit P(A wins) + P(B wins) = 1 most quickly, without algebra? Any 15 year old can calculate that for the</p>
<p>$p = 1/2$ case, $\df...https://math.codidact.com/posts/283366Scalar that varies across one or multiple scalar axesJohnDoeahttps://math.codidact.com/users/363632021-08-13T01:16:31Z2021-08-13T01:16:31Z<p>I have recently read:</p>
<blockquote>
<p>If anything has a quantifiable value that varies across one or multiple scalar axes, it's a signal, and you can study it.</p>
</blockquote>
<p>What ...https://math.codidact.com/posts/283281How do mathematicians measure shape perimeters?JohnDoeahttps://math.codidact.com/users/363632021-08-10T22:40:47Z2021-08-13T00:58:39Z<p>When I create a circle on a computerized plane with a graphic editing program (such as Window's Paint), I can see a group of dots circling around an imagined central point.</p>
<p>if the perime...https://math.codidact.com/posts/283297How to vaticinate calculating $w_{k + 1} = w_1(1 + \dfrac{1- p}{p} + \dots + [\dfrac{1- p}{p}]^k)$ separately for $p = 1 - p$ and $p \neq 1 - p$?DNBhttps://math.codidact.com/users/536282021-08-11T08:35:07Z2021-08-11T08:36:43Z<p>Please see the orange, green, red underlines. 1. Why does $\color{limegreen}{w_{k + 1} = w_1(1 + \dfrac{1- p}{p} + \dots + p\dfrac{1- p}{p}]^k)}$ beget the two cases of $\color{red}p = 1 - p$ a...https://math.codidact.com/posts/283115How does P(Monty opens door 2) = P(Monty opens door 3), and $P(\text{get car}|M_2)P(M_2) = P(\text{get car}|M_3)P(M_3)$?DNBhttps://math.codidact.com/users/536282021-08-05T04:07:10Z2021-08-10T07:20:03Z<p><a href="https://math.codidact.com/posts/283113">"Monty, who knows where the car is, then opens one of the two remaining doors. The door he opens always has a goat behind it (he never reveals th...https://math.codidact.com/posts/283254How to visualize multiplication in the Odds form of Bayes's Theorem?DNBhttps://math.codidact.com/users/536282021-08-10T00:27:13Z2021-08-10T00:28:38Z<p>Here I'm asking solely about the circle pictograms. Please eschew numbers as much as possible. Please explain using solely the circle pictograms. Undeniably, I'm NOT asking about how to multipl...https://math.codidact.com/posts/283253How to visualize division in the Odds form of Bayes's Theorem? DNBhttps://math.codidact.com/users/536282021-08-10T00:04:16Z2021-08-10T00:28:21Z<p>Here I'm asking solely about the circle pictograms. Please eschew referring to, or using, numbers as much as possible. Please explain using solely the circle pictograms. Undeniably, I'm NOT ask...