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NOTES ON METRIC SPACES JUAN PABLO XANDRI 1 Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc We want to endow this set with a metric;I=1 yi b0 P p j=1 bjxij 2 where kkdenotes the Frobenius norm The OLS solution has the form ^b = (X0X) 1X0y Nathaniel E Helwig (U of Minnesota) Multivariate Linear Regression Updated 16Jan17 Slide 11 Multiple Linear Regression ParameterAbout Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How works Test new features Press Copyright Contact us Creators

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P 6 = x=y where x and y are integers with y 6= 0 and gcd( x;y) = 1 Squaring this equation and crossmultiplying we get that 6y 2= x or 23y2 = x2 Therefore, 2 divides x2 = xx Since 2 is prime we must have that 2 divides x Similarly, 3 divides x2 = x x And since 3 is prime we must have that 3 divides x Since 2jx and 3jx and gcd(2;3) = 1, by theIf nis even Properties of Inequalities If aOf x3n L will be added to the first (P − 1) points of x3n We can alternatively view the process of forming the circular convolution x3p n as wrapping the linear convolution x3n around a cylinder of circumference LAs shown in OSB Figure 1, the first (P − 1) points are corrupted by time aliasing, and the points from n = P − 1 ton = L − 1 are identical to the

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Uploaded By DrWasp312 Pages 10 This preview shows page 2 4 out of 10 pages14 (ab) n= a bn 15 a b n = an bn 16 a0 =1wherea2R;a6=0 17 a−n= 1 an;an= 1 a−n 18 ap=q= q p ap 19 If am= anand a6= 1;a6=0then m=n If an= bnwhere n6=0,then a=b 21 If p x;And X n = O p(n−r) if and only if nrX n = O p(1) Proposition 1 if X n and Y n are random variables defined in the same probability space and a n > 0, b n > 0, then (i) If X n = o p(a n) and Y n = o p(b n), we have X nY n = o p(a nb n) X n Y n = o p(max(a n,b n)) X nr = o p(ar n) for r > 0 (ii

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P A n X i 0 P A B i P B i P Y y X k P Y y X k P X k Bayes Rule and with Extra P a n x i 0 p a b i p b i p y y x k p y y x k p x k School Boston University;Gibbs sampling code sampleGibbs4 Duality Theory Recall from Section 1 that the dual to an LP in standard form (P) maximize cTx subject to Ax b, 0 x is the LP (D) minimize bTy subject to ATy c, 0 y Since the problem D is a linear program, it too has a dual

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Yn i=1 p(y ijx i;b 0;b 1;s 2)(1) = i=1 logp(y ijx i;b 0;b 1;s2)(2) = n 2 log2ˇ nlogs 1 2s2 i=1 (y i (b 0 b 1x i)) 2 (3) In the method of maximum likelihood, we pick the parameter values which maximize the likelihood, or, equivalently, maximize the loglikelihood After some calculus (see notes for lecture 5), this gives us the(LMCS,p325) V15 Example Let F(x;y) betheformula x1 ra ndom v ector with mean µ x and v aria nce co v ar iance ma trix !

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F(x)P n(x) dx n =0,1,2,3 Since P n(x) is an even function of x when n is even, and an odd function when n is odd, it follows that if f(x) is an even function of x the coefficients A n will vanish when n is odd;Logs And Exponentials One of the basic properties of numbers is that they may be expressed in exponential form We are all familiar with the representation 1000 = 10 3 or 0001 = 10 3 A more general way of stating this property is to say that any number (N) may be expressed as a base (B) raised to a power (x) or N = B xP yare quadratic surds and if a p x= b p ythen a= band x= y 23 If a;m;nare

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In this problem, let us first look at party since F and G both our solutions to given differential equation, they will satisfy the equation So F dash plus B into F should be equals to kill and jude ash plus be into G equals to kill So let's subtract these two equations We get f trash minus ju dash yes Mhm F minus equals two Q minus Q with zeroS dX t = ( X t ) dt W t e > 0 2 R > d X 0 = x 0 X t = ( x 0 ) e t Z t 0 e ( t s ) dW s a h 1 s E X t jX 0 = x 0 = E ( x 0 ) e t Z t 0 e ( t s ) dW s = ( x 0X y z a b c = Vector equation of any plane that passes through the intersection of planes r n d 1 1= and r n d 2 2= is ( ) ( ) 0r n d r n d1 1− λ − =2 2 , where λis any nonzero constant Cartesian equation of any plane that passes through the intersection of two given planes A 1 x B 1 y C 1 z D 1 = 0 and A

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Given random variables,, , that are defined on a probability space, the joint probability distribution for ,, is a probability distribution that gives the probability that each of ,, falls in any particular range or discrete set of values specified for that variable In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any Electrochemical nitrogen reduction reaction (NRR) is considered as an alternative to the industrial HaberBosch process for NH3 production due to both low energy consumption and environment friendliness However, the major problem of electrochemical NRR is the unsatisfied efficiency and selectivity of electrocatalyst As one group of the cheapest and most abundantNow let's look at the pieces of the new formula X0X = 11 1 X1 X2 2 6 6 6 4 1 X1 1 X2 1 3 7 7 7 5 = n P P Xi Xi P X2 i (X0X)−1 = 1 n P X2 i −( P Xi)2 P X2 i − P Xi − P Xi n 1 nSSX P X2 i − P P Xi i n X0Y = 11 1 X1 X2 2 6 6 6 4 Y1 Y2 Yn 3 7 7 7 5 = P P Yi XiYi Plug these into the equation for b

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Department of Computer Science and Engineering University of Nevada, Reno Reno, NV 557 Email Qipingataolcom Website wwwcseunredu/~yanq I came to the USVar(X) = np(1−p) M(s) = (pes 1−p)n Keeping in the spirit of (1) we denote a binomial n, p rv by X ∼ bin(n,p) 3 geometric distribution with success probability p The number of independent Bernoulli p trials required until the first success yields the geometric rv with pmf p(k) = ˆ p(1−p)k−1, if k ≥ 1;N B a ile y B u x to n Es s e x H e w it S c o ti a L e no i r Wi lt o n Ki t re l U n ic oi Co s ti n Sh i lo h D u bl in D il r d A y d e n A l a r ka S e lm a E f la n d Li b e r t M c Co l l O x fo rd K e lu m S to va ll Ma co n G r ve r Ma rb l e S h e b y N a ki n a C li m a x To p s ai l R wa n E n fie l d S il o a m M ria h H l f x

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Theorem If Xi ∼ exponential(λi), for i = 1,2,,n, and X1,X2,, are mutually independent random variables, then min{X1,X2,,} ∼ exponential i=1 λi ProofThe random variable Xi has cumulative distribution function FX i (x) = P(Xi ≤ x) = 1−e−λ ix i x > 0 for i = 1,2,,n Let the random variable Y = min{X1,X2,,}Then the cumulativeIn particular the variance of b 1 is given by V ar b 1 V ar n X i 1 c i Y i n X from STAT 5102 at The Chinese University of Hong Kong2), p , (x n, y n), let (1110) and S x (1111) y" a n i"1 1y i % y21x i % x2 " a n i"1 x i y i % a a n i"1 x ib a a n i"1 y ib n S x x" a n i"1 1x i % x22" a n i"1 x 2 i % a a n i"1 x ib 2 n yˆ i y i "!ˆ 0 #!ˆ 1x i # e i, i" 1, 2, p , n yˆ "!ˆ 0 #!ˆ 1x JWCL232_c11_qxd 1/15/10 453 PM Page 407 0 y

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Find V (X c )Sho w y our w o rk T hi s is imp orta n t b ecause it tells us w e can a lw a y s pr etend the mea n eq uals ze ro when calculat ing co v aria nce ma trices 6Let X b e a p !Proof write I(X;Y) as a function of p(x) and p(yjx) I(X;Y) = ∑ x;y p(x)p(yjx)log p(yjx) p(y) = = ∑ x;y p(x)p(yjx)logp(yjx) ∑ y {∑ x p(x)p(yjx)} log {∑ x p(yjx)p(x)} (a) Fixing p(yjx), rst linear in p(x), second term concave in p(x) (b) Fixing p(x), complicated in p(yjx) Instead of verify it directly, we will relate it toWhereas if f(x) is an odd function of x, the coefficients A n will vanish when n is even Thus for and even function f(x) we have A

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FANDUBS BR Acho que canto um pouco Não tinha muito pra fazer Resolvi escrever e cantar algumas fandubs Agora estou aqui!Then X/(X Y) ⇠ Beta(a,b)isindependentofX Y ⇠ Gamma(ab,) 11 StudenttIfZ ⇠N(0,1) and Y ⇠ 2 n are independent, then pZ Y/n has the Studentt distribution with n degrees of freedom For n =1,thisbecomesthe Cauchy distribution, which we can also think of as the distribution of Z 1/Z 2 with Z 1,Z 2 iid N(0,1) 4 Sums of3 Independence Xand Y are independent if and only if P(X2A;Y 2B) = P(X2A)P(Y 2B) for all nd B Theorem 2 Let (X;Y) be a bivariate random vector with p

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An bm 1 = bm an Negative exponent "ips" a fraction b0 = 1 b = b1 Don't forget these Convert Radicals to Exponent notation p a = a1=2 m p a = a1=m m p an = an=m Radicals Reducing p a2 b = a p b Remove squares from inside m p am b = a m p b Exponent and Radicals Solving Equations Solve a power by a root xn=m = y ,x = ym=n Solve a root by aCourse Title STAT 110;

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