Recent Questions - Mathematics Stack Exchange |
- Confusing solution to a graph proof problem
- Laplace transform of $t^n$ using integration by parts
- Is the discrete metric equivalent to the d1 metric on R2?
- All ways to understand the word vector
- Sum of independent exponential random variables with common parameter
- Convergence of a series using the ratio test
- Used the definition of convergent proof $z_n$ convergent
- Ask property of gamma distribution
- Prove set equality and function equality
- Is there any relationship between the eigenvalues of $A$ and $AA^T$
- Question about the hierarchical model
- Set Theory Problem (with Symmetric Differences)
- $H^2(\mathrm{im}(f))$ is infinite cyclic for any generic immersion $f\colon S^2\to S^4$
- Prove that the identity map is continuous
- Is the number of distinct characters of a finite group finite?
- Mapping continuously differentiable at a point
- Show that the standard atlas for $\mathbb{RP}^2$, ${(U_i,\phi_i)}_{i=1}^3$, is not orienting.
- Prove that $\arctan(x) = \frac{\pi}{2} - \frac{1}{x} + O\left(\frac{1}{x^3}\right)$ for $x \to +\infty$
- Roots of a trigonometric equation including logarithms
- Is it possible to construct all the $d_i$ based only on the fact that $f(x,y,z)$ is Taylor expandable?
- Why is gravity divided by 2 when you parameterize projectile motion.
- The continuity of the function $F(y)=\displaystyle \int\limits_0^1 \dfrac{yf(x)}{x^2+y^2}dx$
- Why does the inverse limit have two names?
- Intuitive visualization or understanding from $S^2$ to $SO(3)$ and $S^3$ -- and where to find an $S^1$?
- Level set representation of 1-forms in univariate functions
- Expected value and variance of a piecewise function with the integral
- Why probability of unordered samples is not equally likely?
- Probability of being selected in a raffle
- Why does Pearson's chi-squared test divide by the mean and not the variance?
- Limit inferior taken on the norm of a sequence
| Confusing solution to a graph proof problem Posted: 05 Apr 2021 08:12 PM PDT Here's the problem word for word: In a $G=(V,E)$ simple graph $|V|=9, |E|=25$. Show that in $G$ there's at least two vertices with degree of at least $6$. Given solution Proof by contradiction: Let's assume we only have $1$ vertex with a degree of at least $6$, while the rest have a maximum degree of $5$. Since $|E|=\frac{1}{2}\cdot\sum_{i=1}^n d(V_i),\quad |E|=\frac{1\cdot 6 + 8\cdot5}{2}=\frac{46}{2}=23$ However, we need $|E|=25$, therefore we must conclude that $G$ need to have at least two vertices with degree of at least $6$. How i tried it It is possible to construct a graph with the given conditions. Let one vertex have a degree of $10$, and the other eight have a maximum degree of $5$. Then $|E| = \frac{1\cdot 10+8\cdot 5}{2}=\frac{50}{2}=25$. This way it satisfies everything the exercise states. It has the same number of edges and vertices, and it only has one vertex with a degree of at least $6$. Either the problem is worded incorrectly, or i have a poor understanding of graphs. Which one is it? P.S: Yes, the problem text contains the word "least" twice. I checked it.  |
| Laplace transform of $t^n$ using integration by parts Posted: 05 Apr 2021 08:14 PM PDT I am trying to solve the following question:
How would I go about doing this? I am not sure how to manipulate the indefinite integral or use the Laplace transform to get the result. I tried to begin by using the derivative of a Laplace transform, but ended up in a loop. Any guidance is greatly appreciated!  |
| Is the discrete metric equivalent to the d1 metric on R2? Posted: 05 Apr 2021 08:05 PM PDT Is the discrete metric equivalent to the d1 metric (the taxicab metric) on R2? Metrics are defined to be equivalent if we can show that if we have an open ball around point x in one metric, we can find an open ball in the other metric that is contained by the first open ball. In this specific case, I think the metrics cannot be equivalent because for the discrete metric, all open balls less than radius one, we just have the singleton element, but no singleton element is open under the d1 metric. Is the reasoning correct?  |
| All ways to understand the word vector Posted: 05 Apr 2021 08:05 PM PDT I came across the following definitions for the word vector Definition #1 (related to physics)
Definition #2 (related to geometry)
Definition #3 (related to algebra)
Definition #4 (related to vector algebra)
I want to know all the possible definitions (not in rigorous sense) for the word vector  |
| Sum of independent exponential random variables with common parameter Posted: 05 Apr 2021 08:14 PM PDT Let $X$ and $Y$ be independent exponential random random variables with common parameter $\lambda$ and let $Z = X + Y$. Find $f_Z(z)$. My approach: Step 1: $$F_Z(z) = P(X + Y \leq Z) = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{z-x}f_{X,Y}(x,y)dydx$$ Step 2: $$f_Z(z) = \frac{d}{dz}F_Z(z) = \int\limits_{-\infty}^{\infty}\frac{d}{dz}[\int\limits_{-\infty}^{z-x}f_{X,Y}(x,y)dy]dx = \int\limits_{-\infty}^{\infty}f_{X,Y}(x,z-x)dx$$ Step 3: Since the variables are independent: $$f_Z(z) = \int\limits_{-\infty}^{\infty}f_{X}(x)*f_{Y}(z-x)dx$$ Step 4: Using exponential function formula: $\lambda e^{-\lambda x}$ and that lower bound for exponential is 0 to infinity: $$f_Z(z) = \int\limits_{0}^{\infty} \lambda e^{-\lambda x}* \lambda e^{-\lambda (z-x)} dx = \lambda ^2\int\limits_{0}^{\infty} e^{-\lambda z}dx$$ I think I took a long turn somewhere because I'm getting an integral of a constant as my result. Where did I go wrong?  |
| Convergence of a series using the ratio test Posted: 05 Apr 2021 07:57 PM PDT Can you help me with the convergence of the series $$\sum_{n=1}^{\infty} \left( \frac{n}{n+1} \right)^{n(n+1)}$$ I tried the ratio limit but i couldn't obtain the limit, do you have an idea?, the limit of the ratio test is $$\lim_{n \rightarrow \infty} \left(\frac{n+1}{n} \right)^{n(n+1)} \left(\frac{n+1}{n+2}\right)^{(n+1)(n+2)}$$ I review it on Wolframalpha, and it's suppose to converge to $\frac{1}{e}$ but I don't understand why.  |
| Used the definition of convergent proof $z_n$ convergent Posted: 05 Apr 2021 07:48 PM PDT Let $x_{n}$ converges to a and $y_{n}$ converges to b. Then prove that $$z_{n} = \frac{2bx_{n}+2ay_{n} - 2x_{n}y_{n} + ab}{3}$$ Proof that $z_n$ convergent to $ab$  |
| Ask property of gamma distribution Posted: 05 Apr 2021 07:38 PM PDT If $\Sigma X_i \sim gamma(2,\lambda)$, then $\Sigma X_i/\lambda \sim gamma(2,1)$. Why? The background of this question is from George Casella textbook Example 9.2.3.  |
| Prove set equality and function equality Posted: 05 Apr 2021 07:36 PM PDT
Show that if $A$ and $B$ are subsets of $S$ then $A = B \Longleftrightarrow \chi_A = \chi_B$.
($\Longleftarrow$) Assume that $\chi_A = \chi_B$, and let $x \in A$. To show that $x \in B$, it suffices to check that $\chi_B(x) = 1$. But $\chi_B(x) = \chi_A(x) = 1$. (Here the first equality follows from the hypothesis $\chi_A = \chi_B$, and the second equality holds because $x \in A$.) This shows that $A \subseteq B$, and a similar argument shows that $B \subseteq A$. Therefore, $A = B \Longleftrightarrow \chi_A = \chi_B$. QED Is this partial proof acceptable? Can someone suggest a way to prove the other direction?  |
| Is there any relationship between the eigenvalues of $A$ and $AA^T$ Posted: 05 Apr 2021 07:49 PM PDT I am trying to implement some deep learning algorithm based on the derivatives of eigenvalues. The problem that I have is that this line in Pytorch is not implemented which doesn't allow me to calculate the derivatives of complex eigenvalues for real inputs (which is what I want to do)... I have no idea on how to implement this or if it is even possible yet, but a quick solution may be to instead look at the eigenvalues of $AA^T$ which are real. If I could find some relationship between the eigenvalues of $A$ and $AA^T$ I may be able to constrain the eigenvalues of $A$ by using already implemented derivatives for the real eigenvalues of $AA^T$. So is there any relationship between these eigenvalues? I cannot figure out where to search for this kind of information... I plotted histograms of the two eigenvalues and found very different distributions, but IDK where to go from here. Does anyone know?
 |
| Question about the hierarchical model Posted: 05 Apr 2021 07:23 PM PDT I have question about the following problem. Suppose we have data $\left\{X_{i}, i=1, \ldots, n\right\}$ and assume the following hierarchical model. $$ \begin{array}{r} X_{i} \mid \Theta_{i}=\theta_{i} \stackrel{\text { ind }}{\sim} N\left(\theta_{i}, 1\right) \\ \Theta_{1}, \ldots, \Theta_{n} \mid T=\tau \stackrel{\mathrm{iid}}{\sim} N\left(0, \tau^{2}\right) \\ f_{T}(\tau) \propto \frac{1}{\tau} \end{array} $$ The prior on $\tau$ is improper and motivated as a Jeffreys' prior (note that the $N\left(0, \tau^{2}\right)$ belongs to a scale family). Investigate whether the posterior for $\tau$ is a proper density. The following is what I have done: I calculated the joint density of ${X_{i}, {\theta_{i}}}$, and $\tau$, which is: $$\frac{1}{\tau} \prod_{i=1}^{k}\left(\frac{1}{\sqrt{2 \pi}} e^{-\left(X_{i}-\theta_{i}\right)^{2} / 2} \frac{1}{\sqrt{2 \pi \tau^{2}}} e^{-\theta_{i}^{2} /\left(2 \tau^{2}\right)}\right)$$ I can see that theoretically we could integrate out all $\theta_{i}$ although it seems cumbersome since we have these exponentials with $\theta_{i}$ 's in them, but would we not get then the joint density of $\tau, X_{1}, \ldots, X_{k},$ whereas we want the conditional density of $\tau$ on these $X_{i}$ 's?  |
| Set Theory Problem (with Symmetric Differences) Posted: 05 Apr 2021 08:09 PM PDT I'm reading "Shattering-extremal set systems of VC dimension at most 2" by Tamás Mészáros and Lajos Rónyai, which is not about set theory, but uses a lot of set theory and shows none of its steps. I've been left to fill in the blanks, which I'm really struggling with! I'm trying to draw lots of Venn diagrams but I can't seem to get anywhere. Here is the information: $A\bigtriangleup B=\{x_1\},B\bigtriangleup C =\{x_2\}, D=B\bigtriangleup \{x_1,x_2\}, C\bigtriangleup D=\{x_1\},A\bigtriangleup D=\{x_2\}.$ They claim that the following sets will each be one of $\{A,B,C,D\}$: $$B\cap D, (B\cap D)\cup \{x_1\},(B\cap D)\cup \{x_2\}, (B\cap D)\cup \{x_1,x_2\}.$$ I'm trying to use the definition of the symmetric difference, but my expressions get very messy very fast. Mészáros, Tamás; Rónyai, Lajos, Shattering-extremal set systems of VC dimension at most 2, Electron. J. Comb. 21, No. 4, Research Paper P4.30, 17 p. (2014). ZBL1302.05201.  |
| $H^2(\mathrm{im}(f))$ is infinite cyclic for any generic immersion $f\colon S^2\to S^4$ Posted: 05 Apr 2021 07:21 PM PDT Just to give the context: this appeared while defining an invariant for a 2-dimensional link map $(f_1,f_2)\colon S^2\coprod S^2\to S^4$ (in "The Group of Disjoint 2-Spheres in 4-Space" by Schneiderman and Teichner.) Specifically, they deform the link map into a generic immersion, and use the Hurewicz maps, Alexander duality and the title to get $$ \pi_1(S^4\setminus\mathrm{im}(f_i))\twoheadrightarrow H_1(S^4\setminus\mathrm{im}(f_i))\cong H^2(\mathrm{im}(f_i))\cong\mathbb{Z},$$ and thus are able to consider Wall's intersection number $\lambda(f_i,f_i)$ with values in the group ring $\mathbb{Z}[\mathbb{Z}]$. I have no clue on computing the homology of the image of generic immersions (or any continuous maps,) so any idea or suggestion would be very helpful.  |
| Prove that the identity map is continuous Posted: 05 Apr 2021 07:21 PM PDT Hello I have problems with exercise Let $\tau_1$ and $\tau_2$ be given topologies on a set $X$. Prove that the identity map $(X, \tau_1) → (X, \tau_2)$, $x \longmapsto x$, is continuous if and only if $\tau_1$ is finer than $\tau_2$ My attempt: $\Rightarrow{}$) We have the function is continuous then $\tau_1 \subseteq{\tau_2}$ $\therefore$ $\tau_1$ is finer than $\tau_2$ $\Leftarrow{}$) By hypothesis $\tau_1$ is finer than $\tau_2$ then $\tau_1 \subseteq{\tau_2}$. Le the function defined by $(X, \tau_1) → (X, \tau_2)$, $x \longmapsto x$ I don't know how to continue Thanks  |
| Is the number of distinct characters of a finite group finite? Posted: 05 Apr 2021 07:22 PM PDT Let $G$ be a finite group, $V$ a finite complex vector space and $\rho: G \to GL(V)$ be a representation of $G$ in $V$. The character $\chi_{\rho}$ associated to $\rho$ is a mapping $\chi_{\rho}: G \to \mathbb{C}^{\times}$ defined by: $$\chi_{\rho}(g) := \text{Tr}(\rho(g))$$ It is an important result of the theory of representations of finite groups that two isomorphic representations give the same character. Question: Is the number of distinct irreducible characters of a finite group $G$ finite?  |
| Mapping continuously differentiable at a point Posted: 05 Apr 2021 07:39 PM PDT I am reading Coleman's Calculus on Normed Vector Spaces. He defines what it means for a mapping $f: O \to F$ from an open subset $O$ of a normed vector space $E$ to a normed vector space $F$ to be differentiable at a point $x \in O$. Also, such a mapping is differentiable if it is differentiable at every point in its domain. He than defines that such a mapping is continuously differentiable (or of class $C^1$) when it is differentiable and its differential mapping $f': O \to \mathcal{L}(E,F)$ is continuous. He does not define what it means for a mapping to be continuously differentiable at a point (at least I could not find a definition), so I am wondering what such definition would be like. This is what I first thought of: A mapping $f: O \to F$ is continuously differentiable at a point $x \in O$ if there is a neighborhood $N$ of $x$ such that $f$ is differentiable at $N$ and $f': N \to \mathcal{L}(E,F)$ is continuous at $x$. My questions are:
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| Show that the standard atlas for $\mathbb{RP}^2$, ${(U_i,\phi_i)}_{i=1}^3$, is not orienting. Posted: 05 Apr 2021 08:08 PM PDT Show that the standard atlas for $\mathbb{RP}^2$, ${(U_i,\phi_i)}_{i=1}^3$, is not orienting. $U_1=\{[x^1,x^2,x^3]\in \mathbb{RP}^2 \ | \ x_1 \neq 0\}$, $\phi_1 ([x^1,x^2,x^3])=(\frac{x^2}{x^1},\frac{x^3}{x^1})$ My attempt: An atlas $\mathcal{A}$ is orienting if any two charts in $\mathcal{A}$ are orientation compatible. It suffices to prove that there are 2 charts that are not orientation compatible in the standard atlas for $\mathbb{RP}^2$. Let's check the compatibility of the charts $(U_1,\phi_1)$ and $(U_3,\phi_3)$ $\phi_3 \circ (\phi_1)^{-1} (x^1,x^2)=\phi_3 ([1,x^1,x^2])=(\frac{1}{x^2},\frac{x^1}{x^2})$ $[(\phi_3 \circ (\phi_1 )^{-1})_*]= \begin{bmatrix} 0 & \frac{-1}{(x^2)^2} \\ \frac{1}{x^2} & \frac{-x^1}{(x^2)^2} \end{bmatrix} $ $\text{det} [(\phi_3 \circ (\phi_1 )^{-1})_*] =\frac{1}{(x^2)^3}$ I could not decide weather this is positive or negative in $\phi_1 (U_1 \cap U_3)$. I know that $U_1 \cap U_3=\{[x^1,x^2,x^3] \ | \ x^1,x^3\neq 0 \}$ $\phi_1 (U_1 \cap U_3)=\{(\frac{x^2}{x^1},\frac{x^3}{x^1}) \ | \ x^1,x^3\neq 0 \}=\mathbb{R}^2 \setminus \{y=0\}$ But still cannot figure it out. I also calculated $\text{det} [(\phi_2 \circ (\phi_1 )^{-1})_*] =\frac{-1}{(x^1)^3}$, and $\text{det} [(\phi_3 \circ (\phi_2 )^{-1})_*] =\frac{-1}{(x^2)^3}$. I still do not know weather they're positive or negative. Any help would be appreciated.  |
| Posted: 05 Apr 2021 07:18 PM PDT Want to prove that $$\arctan(x) = \frac{\pi}{2} - \frac{1}{x} + O\left(\frac{1}{x^3}\right) \ \text{as}~~~ x \to +\infty$$ Wolfram says that it's true but I'm trying to find some formal prove of this equality. Here's what i found about that: We know that $\arctan(x) = \frac{\pi}{2} - \int_x^{+\infty}\frac{dt}{1 + t^2}$ (we can easily prove this by solving such integral). But what this one can give us for asymptotic assessment? Seems like we have to prove that this integral equals to $-\frac{1}{x} + O\left(\frac{1}{x^3}\right)$, but how?  |
| Roots of a trigonometric equation including logarithms Posted: 05 Apr 2021 08:10 PM PDT How would you solve this equation? Is it only possible to approximate the roots, if any? $x\ln(x)\cos(x) + \sin(x) = 0$ My attempt: Im unsure how to proceed from here, and if whether treating $xlnx$ as a constant is possible.  |
| Posted: 05 Apr 2021 08:02 PM PDT If we have a function of three variables $x,y,$ and $z$ in the following form $$f(x,y,z)=\frac{e^{[x+y+z]}}{d_1d_2}+\frac{e^{[x-y-z]}}{d_2d_3}+\frac{e^{[-x-y+z]}}{d_3d_1}$$ where $d_1$, $d_2$, and $d_3$ are linear combinations of $x$, $y$, and $z$, is it possible to construct all the $d_i$ based only on the fact that $f(x,y,z)$ is Taylor expandable? The final solution should look like: $$f(x,y,z)=\frac{e^{[x+y+z]}}{(x+y)(y+z)}-\frac{e^{[x-y-z]}}{(x-z)(y+z)}+\frac{e^{[-x-y+z]}}{(x+y)(x-z)}$$ but I cannot see how we can find the denominators.  |
| Why is gravity divided by 2 when you parameterize projectile motion. Posted: 05 Apr 2021 07:23 PM PDT I'm currently working on a pretty basic problem about projectile motion. No air friction, just gravity. There is a ball that is being launched from $4$ feet up at $80$ ft/sec at $40$ degrees above the horizontal. Gravity is assumed to be $32$ feet/$\text{sec}^2$. I need to parameterize it. I got $$x = 80\cos(40)$$ and $$y = 4 + t(51.4-32t) $$ My question is, when I check my answer with an online solver, why does it say that the coeffecient in the second equation, $32$, should be $16$? Why is gravity divided by $2$? Thanks for helping.  |
| The continuity of the function $F(y)=\displaystyle \int\limits_0^1 \dfrac{yf(x)}{x^2+y^2}dx$ Posted: 05 Apr 2021 08:04 PM PDT Considering the continuity of the function $F(y)=\displaystyle \int\limits_0^1 \dfrac{yf(x)}{x^2+y^2}\,dx$, with $f(x)$ is a continuous function on $[0;1]$. I will consider $y>0, \, y<0 \, \text{and} \, y=0$. 1. Consider $y=y_0>0$. Then, $\exists \, c, \, d: 0<c<y_0<d$. And, we have $\dfrac{yf(x)}{x^2+y^2}$ is continuous on $D: 0 \le x \le 1, \, c \le y\le d$. So, $F(y)$ is continuous on $[c;d]$, or $F(y)$ is continuous at the point $y=y_0$. 2. Consider $y=y_0<0$. Proof is as same as above. 3. Consider $y=0$. I think it isn't continuous at this point. However, I don't know how to prove it  |
| Why does the inverse limit have two names? Posted: 05 Apr 2021 07:33 PM PDT Why do we call the inverse limit a projective limit? If I've understood correctly there are two name for the same object so I would glad to know if there is a simple explanation of this.  |
| Posted: 05 Apr 2021 07:23 PM PDT We know that
My questions are that -
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| Level set representation of 1-forms in univariate functions Posted: 05 Apr 2021 07:20 PM PDT This question is in connection to an excellent youtube playlist by @eigenchris on Tensor Calculus. The idea of a differential $\mathrm d$ operator as turning a scalar field $f$ ($0$-form) into a covector field $\mathrm df$ ($1$-form) is nicely explained in this presentation, based on contour maps of a function $f(x,y).$ The scalar field is the value of $f$ for every pair in $(x,y),$ and the covector field is represented by the level set curves. Given a vector at any point, the covector field would produce the directional derivative, $\mathrm df(\vec v).$ Similarly the actual $x$ and $y$ independent variables are scalar fields with each point attached to their own values:
The explanation is clear within the confines of a multivariable function, although clearly the level sets representing $\mathrm dx$ and $\mathrm dy$ on the right column of the plot above are completely random vertical and horizontal lines. In Wikipedia a scalar field is defined as
and a region as
Presumably, then, the construct applied to a unvariable function $f(x)$ would be as follows: The scalar field $x$ is simply the $x$ axis with $\mathrm dx$ best represented by a homogeneously changing color gradient on a line from blue to red:
understanding the real line (or the domain of the function on the real line) as the region $U$ (?). Here $y$ would not be a smooth color gradient as in the graph above, because it will be dependent on $x$ through the action of $f.$ The question is:
Would it make sense to picture it like this...
in the case of
for example?
For example, the function $f(x) = x^3,$ which in the interval $[1,10],$ has as many level sets as real numbers in the interval $[1,10].$ Therefore the idea of "counting the level sets pierced by a vector" when integrating from $1$ to $10,$ as suggested on this slide on the same series of presentations:
would presumably translate into $\color{red}{f(x)dx=x^3 dx},$ with the number of counted level sets pierced by the arrow being infinite, as we can clearly increase the number of level sets represented in the diagram below (from $5$ to $10$ to $20$ from left to right):
The actual values at each level set would eventually reproduce the entire $y=x^3$ function in that segment, and multiplying each value, $\color{red}{x^3},$ times a small interval of $x,$ i.e $\color{red}{dx},$ would indeed bring us back to the Riemann integral. But that clearly defeats the idea of just counting the level sets pierced by the arrow.
The $\int_1^{10}x^3 dx \approx 2499,$ while adding the level set lines on the plot to the left results in crossing $6$ level sets, each separate by $166.5,$ which would result in $6\times 166.5=999.$ Wrap up after EDIT of @eigenchris answer:
How do you go from here to $2499.75$ piercing the level sets? In the plot the level sets are separated by $100$ and there are $25$ level sets, resulting in $100 \times 25 = 2500.$ Since we started at $F(1)=\frac 1 4 (1)^4 =\frac 1 4,$ we end up with $2500 - \frac 1 4 = 2499.75.$  |
| Expected value and variance of a piecewise function with the integral Posted: 05 Apr 2021 08:01 PM PDT I have the following stepwise function: $$f(x) = \begin{cases} \frac{1}{3} & -1\le x \lt 0 \\ \frac{2}{3} & 0 \le x \le 1 \\ 0 & \text{otherwise} \end{cases} $$ I wonder how I can derive $E(X)$ and $Var(X)$ of a stepwise uniform function using the integral. If I draw it I would say the mean is at $x = 1/4$ (as then the area below the function equals $0.5$). But when I am integrating it I get $1/6$. Can someone help me and show how I properly solve this question? Thanks a lot!  |
| Why probability of unordered samples is not equally likely? Posted: 05 Apr 2021 07:32 PM PDT Consider a survey where a sample of size k is collected by choosing people from a population of size n one at a time, with replacement and with equal probabilities. Then the n^k ordered samples are equally likely, making the naive definition applicable, but the "n+k-1 choose k" unordered samples (where all that matters is how many times each person was sampled) are not equally likely. [Source: Introduction to probability by joseph k. blitzstein] Could you please explain why is it like that?  |
| Probability of being selected in a raffle Posted: 05 Apr 2021 07:25 PM PDT There's a raffle with 1,000 names in a bucket. 600 of those names are in there once, and 200 are in there twice. So, just to reiterate, there are 800 unique names in the raffle, and 1000 names total. 500/1000 names will be selected from the raffle. Q1: What is the probability of being selected if your name is in the bucket once? Q2: What is the probability of being selected if your name is in the bucket twice? The answers are: Q1: 500/100 = 50% chance of being drawn if your name is in the bucket once Q2: 500/1000 + (500/1000)*(499/999) = 75% chance of being drawn if your name is in the bucket twice. How do these answers make sense? Is there another way of arriving at these answers? I can't seem to understand these conceptually. Thank you!  |
| Why does Pearson's chi-squared test divide by the mean and not the variance? Posted: 05 Apr 2021 07:26 PM PDT I am wondering why in Pearson's chi-squared test, the divisor of each element in the sum is the matching expectation and not the matching variance. As I understand it, the test works by standardizing each normal variable before summing, so the results set can be tested against the chi-squared distribution which deals with a sum of squares of standard normal random variables. The way a normal random variable is standardized is by subtracting the expectation and dividing by the standard deviation. So, in Pearson's test, this should give the variance in the divisor of each element, not the expectation.  |
| Limit inferior taken on the norm of a sequence Posted: 05 Apr 2021 07:36 PM PDT Let $E$ a normed vector space and let $(x_n)$ be a sequence in $E$. Suppose that $x_n$ converges weakly (i.e. wrt the weak topology) to $x$. Why is it that from the inequality $$ |f(x_n)| \leq \|f\| \|x_n\|, $$ passing to the limit we obtain $$ |f(x)| \leq \|f\| \lim\inf\|x_n\| $$ ? Particularly, why can't we simply write $\lim \|x_n\|$ ?  |
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