I'm reading a .pdf file about Probability.  the link  The author introduces the lemma 4.1, suggesting to consult the proof in a specific book. I tried to search the lemma in the book but i didn't find it. Please help me to find a name for this lemma so as i can consult its proof...     Lemma 4.1. Let $Z_1, Z_2, \dots$ be a sequence of random variables having distribution functions $F_{Z_n}$ and moment generating functions $M_{Z_n}$ s.t. $n \geq 1$. Furthermore, let $Z$ be a random variable having distribution $F_Z$ and moment generating function $M_Z$. If $M_{Z_n}(t) \to M_Z$ for all $t$, then $F_{Z_n}(t) \to F_{Z}(t)$ for all $t$ at which $F_Z(t)$ is continuous.
I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ?
When I upload an image to Stack Overflow, I cannot see my JPEG file in my post. It's all blank. I've tried it with different files, but gotten the same result. I hope I'm right in thinking this is the site where you post general computer-related problems...  A problem I have when playing games (since you won't be pressing multiple keys simultaneously anytime else), is that when I press up/left and then another key, it fails to 'register' as if that last key is never pressed. This doesn't happen when I'm pressing any other two arrow keys, e.g. down/left or up/right. The weird thing is that when I'm playing with WASD instead of the arrow keys, i still have this problem! Meaning W+A+another button doesn't work.  If anyone knows what I should, I'd appreciate the help. (My reflexes are fast and I can always circumvent this problem, but it's quite annoying and shouldn't exist to begin with!) What does "calculus" mean? "calculus" and "formal system"  From      a calculus is a formal system that consists of          a set of syntactic expressions (well-formed formulæ or wffs),    a distinguished subset of these expressions (axioms), plus    a set of formal rules that define a specific binary relation on the space of expressions.      From      Formal systems in mathematics consist of the following elements:         A finite set of symbols (i.e. the alphabet), that can be used for constructing formulas (i.e. finite strings of symbols).   A grammar, which tells how well-formed formulas (abbreviated wff) are constructed out of the symbols in the alphabet. It is   usually   required that there be a decision procedure for deciding whether a   formula is well formed or not.   A set of axioms or axiom schemata: each axiom must be a wff.   A set of inference rules.      what is the difference between a formal system, and a calculus, then? I think they are the same? "calculus" and "logic system"  "Calculus" appears in "propositional calculus" and "first-order predicate calculus", which are also called "propositional logic" and "first-order logic" respectively. So I thought "calculus" and "logic" mean the same, and "a logic" is, according to ,      A logical system or, for short, logic, is a formal system together with a form of semantics, usually in the form of model-theoretic   interpretation, which assigns truth values to sentences of the   formal language, that is, formulae that contain no free variables.    But then I saw "calculus" also appears in "lambda calculus", which is also a formal system. I think a lambda calculus isn't a logic system, right? What does "calculus" mean in "lambda calculus"? Furthermore  "calculus" can also mean computational real analysis for first-year college students.   Thanks. Convert from LyX/Latex to wordpress post I have some documents in LyX/Latex and I want them as a wordpress post. I have tried converting latex to HTML (with latex2html) and afterwards, pasting that code in the wordpress editor but everything looks ugly. Do you know a better approach? So earlier today I came across , and I am having some trouble understanding the solution. The question is as follows:     You throw a fair six-sided die until you get 6. What is the expected   number of throws (including the throw giving 6) conditioned on the event   that all throws gave even numbers?   Quoted from Jimmy Jin in "Elchanan Mossel’s dice problem"  In the paper it goes on to state why a common wrong answer is $3$. Then afterwards explains that this problem has the same answer to,   "What is the expected number of times you can roll only $2$’s or $4$’s until you roll any other number?"  I don't understand why this is the case. If the original problem is asking for specifically a $6$, shouldn't that limit many of the possible sequences?  I also attempted to solve the problem using another method, but got an answer different from both $3$ and the correct answer of $1.5$.  I saw that possible sequences could have been something like:  $$\{6\}$$ $$\{2,6\}, \{4,6\}$$ $$\{2,2,6\}, \{2,4,6\}, \{4,2,6\}, \{4,4,6\}$$ $$\vdots$$  To which I set up the following summation and solved using :  $$\text{Expected Value} =\sum_{n=1}^\infty n\left( {\frac{1}{6}} \right)^n 2^{n-1} = 0.375$$ Obviously this is different and probably incorrect, but I can't figure out where the error in the thought process is.  Any help on understanding this would be greatly appreciated.  A blog post discussing the problem can be found Let $A, B \in \mathcal M_n(\mathbb C)$, we want to know if there's any relation between their spectral radius and the spectral radius of $A + B$.  The first part of this exercise is giving examples of the three posibilities: $\rho(A) + \rho(B) &lt; \rho(A+B)$ ; $\rho(A) + \rho(B) &gt; \rho(A+B)$ and $\rho(A)+ \rho(B) =\rho(A+B)$. This part was easy because there are no conditions about A or B.  The second part is proving that if A and B are normal matrixes, (i.e. $ AA^* =A^*A$, where $A^*$ is its conjugate transpose), then $ \rho(A+B) \le \rho(A) + \rho(B) $.   So my first attempt was using the Schur's decomposition, so $A=U^{-1}D_AU$ where $D_A$ is a diagonal matrix with the eigenvalues of $A$ in its diagonal and U is an unitary matrix. Similarly, B is similar to a diagonal matrix $D_B$ with the eigenvalues of B in its diagonal. That way I have $\rho(A) = \rho(D_A)$ and $\rho(B) = \rho(D_B)$ so now since $D_A + D_B$ is a diagonal matrix with $\lambda_i + \mu_i$ for $ 1\le i \le n $, where $\lambda_i$ and $\mu_i$ are the eigenvalues of $A$ and $B$ respectively. From here it's easy to see that $ \rho(D_A+D_B) \le \rho(D_A) + \rho(D_B) $, since $ |\lambda_i+\mu_i|\le|\lambda_i|+|\mu_i| \ \forall i$. But I don't see how to include $\rho(A+B)$ in this.  And the last part of the exercise is proving that if $ \{X_i\}$ is a basis of eigenvectors, which is common to A and B, then $ \rho(A+B) \le \rho(A) + \rho(B) $.  Here's my attempt, but I would like to know if it's correct:  Let $\lambda_i$ and $\mu_i$ be eigenvalues of A and B associated to the vector $X_i$, then $AX_i=\lambda_i X_i$ and $BX_i = \mu_iX_i$, and if we sum both equalities, we have $AX_i + BX_i = \lambda_iX_i + \mu_iXi$ so $(A+B)X_i=(\lambda_i+\mu_i)X_i$ so $(\lambda_i+\mu_i)$ is an eigenvalor of $(A+B)$ associated to the eigenvector $X_i$. Then is again clear that $|\lambda_i+\mu_i|\le|\lambda_i|+|\mu_i| \le \rho(A) + \rho(B)\ \forall i$, and particularly, for $\max_i |\lambda_i+\mu_i| = \rho(A+B)$ so $ \rho(A+B)\le \rho(A) + \rho(B)$  Any kind of help is appreciated and thank you very much in advance! How do I increase the render distance? What is the difference between linear regression and logistic regression? What is the difference between linear regression and logistic regression?  When would you use each? Show that the set of all polynomials whose constant term is a multiple of 3 is an ideal of the ring $Z[x]$.   Attempt: Suppose then we need to show   if $p(x), q(x) \in I$ then $p(x) + q(x) \in I$  and if $r(x) \in Z[x]$ , then $p(x)r(x) \in I$ and $r(x)p(x) \in I$.  And $0 \in Z$, then $0 \in I$.  Then let   $I =$ {$a_nx^n + a_{n-1}x^{n-1} + ...+a_1x + a_0$ | $a_i \in Z$, $a_0\in 3Z$}  $p(x)  = a_nx^n + a_{n-1}x^{n-1} + ...+a_1x + 3a$  and $q(x) = b_mx^n + b_{m-1}x^{n-1} + ...+b_1x + 3b$  Here we do not assume that $p(x) $ and $q(x)$ have the same degree.  Then $p(x) + q(x)  = [a_nx^n + a_{n-1}x^{n-1} + ...+a_1x + 3a]  +  [b_mx^n + b_{m-1}x^{n-1} + ...+b_1x + 3b] = 3(a + b) + (a_1 + b_1)x + ....+ (a_n + b_m)x^n$.  so $p(x) + q(x) \in I$  And $0 \in Z$ is also in $I$, since for $p(x) = 0x^n + ....+ 3(0) = 0$.  And let $r(x)= c_nx^n + ...+3c  \in Z[x] $.  Then $r(x)p(x) = 3(3ac) + (3b_1 + 3a_1) + ....+ a_nb_mx^{2n}$  So $r(x)p(x) \in I$.  Can someone please help? I am not really sure how to prove this. I am getting confused with the underscrips Any help or better approach would really help. Thanks in advance! How do I resize my current ubuntu partition? I've got Ubuntu 10.10 installed to use my whole hard-disk, but would like to use the 130GB of free space I've got to create a new partition and do some development on Unity and Natty.  Can I use gparted to do this?  If so how? I've recently run into many questions where to author asks several related questions in one post, many of them asked before.   I'm wondering what out policy is on such questions:   Should I vote-close them as being not constructive? Should I vote-close them as duplicates? Should I leave them be? Or even suggest an answer?    - Each of the questions: Waiting for an AJAX request with selenium, and headless browsing in selenium have been asked here before, but he asked both in one.  I've done my best to search for similar questions here in meta but I was unable to find any, reading appreciated. Financial proof to get a Schengen visa to France I am planning to apply for a Schengen visa to visit France. I have an attestation d'acceuil, but the visa requires I submit bank statement for the past 3 months. I can get this statement, but the problem is that my company has been paying my salary in cash for the past months, and I place the money into the account by myself.  Can I submit the following?   my accounting book (current account)  letter from the bank stating that I have money in it payslips for the past 3 months   Will that satisfy the financial requirements for the visa? Why doesn't my Bash script recognize aliases? I need to get MODIS NDVI (MOD13A3), but output from EarthExplorer or Daac2Disk HDF format is a numerical range is -2000 to 9996.   But I need for futher work basic range -1 to 1.   Is there any conversion method? He called her, emailed her, texted her, tweeted her—all to no use.   Strictly speaking, I would need to write texted her and tweeted her, but I'm omitting and to convey a rhythm and sense of urgency. What do you call this kind of construction, and is it accepted grammar (at least for creative writing, if not in an academic context)? I recently upgraded to Ubuntu 13.10, with the new keyboard handling system. The little "switch keyboard" option does not let me set it to Alt+Shift. How can I get this keyboard binding, as I have absolutely no interest in using the Mac-esque Super+Space combo? siunitx tabular numbers with separated uncertainty I would like to format the space reserved for a number shown in a table with separated uncertainty. It seems that when I use the table-format option the brackets around the mantissa disappear. This is illustrated in the example below:  \documentclass{article}     \usepackage{siunitx}  \begin{document}     \begin{table}   \begin{tabular}{                   S[separate-uncertainty = true]                   S[table-format=1.1(1)e1,                   separate-uncertainty = true]                   }     {Header} &amp; {Header}\\     1.0(2)e2 &amp; 1.0(2)e2\\   \end{tabular} \end{table}     \end{document}     I have also tried to use the bracket-numbers and multi-part-units = brackets without solving this issue. Is there any workaround to center the number and keep the brackets like in the first column of the example? I used to deploy metadata using ant migration tool successfully but from last night I am getting the below error:  John@John:~/mm_workspace/TEST_Project/deploy$ ant deployUnpackaged  Buildfile: /home/john/mm_workspace/TEST_Project/deploy/build.xml  deployUnpackaged:  BUILD FAILED /home/john/mm_workspace/TEST_Project/deploy/build.xml:72: Failed to submit request to https://cs19.salesforce.com/services/Soap/m/35.0/00D293456000Qaf for user john@test.com.dev. Error: {http://schemas.xmlsoap.org/soap/envelope/}Client - Maximum size of request reached. Maximum size of request is 52428800 bytes.  Total time: 2 minutes 10 seconds   I think this is because of too much metadata that I am trying to deploy. For now, I am deploying classes and then VF pages and so on but I am looking for an option to deploy everything in one go without running into the above error.  I am sure that people are working on more complex projects than mine with lots of metadata and I am wondering how are they deploying with out hitting this error?  Thanks Android: textview hyperlink match first column of file a with paragraphs of file b I have 2 files. The first, fileA looks like  TCONS_00000066  XLOC_000030     -       u       q1:XLOC_000030|TCONS_00000066|0|0.000000|0.000000|0.000000|0.000000|- TCONS_00000130  XLOC_000057     -       u       q1:XLOC_000057|TCONS_00000130|0|0.000000|0.000000|0.000000|0.000000|- TCONS_00000395  XLOC_000206     -       u       q1:XLOC_000204|TCONS_00000393|0|0.000000|0.000000|0.000000|0.000000|-   FileB looks like:  &gt;TCONS_00000001 gene=XLOC_000001 AGATGAGCTGGTGGGGATGCTCTAAGAGAACGAGAGAAGCACAGAGCAGATAAACCACACCCACAGGCAC CACCGTCCTTGTTGGTAATGAAGAAGACGAGACGACGACTTCCCCACTAGGAAACACGACGGAGGCGGAG ATGATCGACGGCGGAGAGAGCTACAGAAACATCGATGCCTCCTGTCCAATCCCCCCATCCCATTCGGTAG TTGGATTGAAGACTACCGAATAAGAGAAGCAGGCAGGCAGACAAACCCTTGAACCAAGGAGTCCTCGCTG AGGAAGCTTTGGATCCACGACGCAGCTATGGCCTCCCCGCCCACCAGGCCGCCAGCCACAACCAGCTGAC TAGGTCGCATGCATCATCAGATTTCAATCTCCCTTCGTTCCCTGTCCCTAATCCAATACCAATAGGGAGC AATCAGCTGCTCCTCGACGGCGAGGGAGATGTCGTCGGCCGCGGGCCAAGACAACGGAGATACCGCTGGG GACTACATCAAGTGGATGTGCGGCGCCGGTGGCCGTGCGGGCGGCGCCATGGCCAACCTCCAGCGCGGCG TTGGCTCCCTCGTCCGTGACATTGGCGACCCCTGCCTCAACCCATCCCCCGTTAAGGGGAGCAAAATGCT CAAACCGGAAAAATGGCACACATGTTTTGATAATGATGGAAAGGTCATAGGTTTCCGTAAAGCCCTAAAA TTCATTGTCTTAGGGGGTGTGGATCCCACTATTCGAGCTGAAGTTTGGGAATTTCTTCTTGGCTGCTATG CCTTGAGTAGTACCTCAGAGTATAGGAGGAAACTAAGAGCTGTTAGAAGGGAAAAATATCAAATTTTAGT TAGACAGTGCCAGAGCATGCACCCAAGCATTGGTACAGGTGAGCTTGCTTACGCTGTTGGATCAAAGCTA   Now, fileA contains selected transcript numbers in the first column and fileB contains sequences of all transcripts. I want to scan fileB for the first column of fileA and print the trailing sequences of matching transcripts along with the transcript number.