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Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, history, ) = (x )2/2 2 2 2 µ σ πσ µσ • The notation N(µ, σ2) means normally distributed with mean µ and variance σ2 If1(xµ)(1−p)f 2(xµ,τ2), where f 1(xµ) = 1 √ 2π exp ˆ − (x−µ)2 2 ˙ is the N(µ,1) density, and f 2(xµ,τ) = 1 √ 2πτ2 exp ˆ − (x−µ)2 2τ2 ˙ is the N(µ,τ2) density Then the expectation of a random variable with this mixture density is given by EX i = Z ∞ −∞ xf(xµ,τ2,p)dx = Z ∞ −∞ x pf 1(xµ

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} o p } w } µ o ^ v d p µ o h v d v } w p p % (vfuleh xqd 9 vl od dilupdflyq hq yhugdghud \ ) vl od dilupdflyq hv idovd 9 /rv pd\dv frqvwux\hurq judqghv flxgdghv frpr 7lndo \ 3dohqtxh ) (o lqwhufdpelr gh surgxfwrv hud vror xqd dfwlylgdg vlq3 If c is a v ec tor of constan ts, X c !So 131 and 433 (3 sf) σ µ So the probability that at least 15 table cloths can be sold is dp ) (© Pearson Education Ltd 19 Copying permitted for purchasing institution only This material is not copyright free 7 13 a b 1 18

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Morgan , West Virginia Personal income last published on November 14, 19 Morgan is one of 55 counties in West Virginia It is part of the HagerstownMartinsburg, MDµ µ µ ξµ ξ ξ = ∑∫ (356) where the kernel K is calculable in pQCD (for µ2 and µ 0 2 large) We can think of this relation as describing the evolution of a parton of type i and scale µ2 to a parton of type j at a scale µ0 2 (µ 0 2 < µ2) ASIDE Note that, having onceD ô ì ß

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For example, we denote the population mean by µ, and we can use the sample mean x¯ to estimate µ Suppose we wanted to know the average income of households in NC To estimate this population mean income µ, we may randomly take a sample of 1000 households and compute their average income x¯ and use this as an estimate for µº m x P U d!!!• Two parameters, µ and σ Note that the normal distribution is actually a family of distributions, since µ and σ determine the shape of the distribution • The rule for a normal density function is e 2 1 f(x;

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¡gdx f dy = Z Z R µ @f @x @g @y ¶ dA Note @f @x @g @y is the twodimensional divergence of the vector fleld F = hf;gi If the divergence of F = hf;gi is 0 throughout a region R the vector fleld F is said to be sourcefree Example Use Green's Theorem to compute the °ux integral (outward) of the radial vector fleld F = hx;yiÐ D ( p!!W ð } ( Z > v } µ } v t v Ç ï í D Z U ó X ï ì u t õ u U o Ç Z À ^ u W Z W l l µ ì î Á X Ì } } u X µ l i l ô ô ñ ï ñ ò ó ì ò ï ó M Á AYt Æ t,Zt s ðY i id, D^t o Ì ð Ì ì õ

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D Ø b ½ @!!!Ù å ) Å B p ì î!!• Nuclear shadowing suppression fA(x,µ2) < A fN(x,µ2) for small x < 0005 • Important for QCD phenomenology of hard processes with nuclei cold nuclear matter effects, gluon saturation (RHIC, LHC, EIC, LHeC/FCC) • fA(x,µ2) are determined from global QCD fits to data on fixedtarget DIS, hard processes in dA (RHIC) and pA (LHC) Q → fA(x,µ2) with significant uncertainties

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Oct 28, 04 · It is valid as long as f and g have the same domains and ranges Feb 28, 08 #5 JasonRox Homework Helper Gold Member 2,314 3 ObsessiveMathsFreak said This appears to be the definition of the sum of two functions It is valid as long as f and g have the same domains and ranges They don't need the same range) 4 If A is a ma trix o f consta n ts, AX !ú b V z!!

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T Z À µ Z } u v Ç v P ( v v o u v } ( } ( Z D P Z Ç & } } v l U / v X ~ Z ^& } } v l _ U Á Z Z } u Z u v } ( ( v v o } } v } ( ^ u ï ì U î ì í õ U v ZAnswer to d Compute dif(g(x)), where f(x) = x'and g(x) = 9x6 d To compute f(g(x)), first find f'(x) and g'(x) f'(x) = g'(x) =(Fg) (x) =F (x) g (x) Answer it is a operation of functions in multiplication actually there are 4 operations

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ï v ( U î í v } ( } µ r } ( r } l v v P v î ì í ò Á v } } v µ P XVertical Translation For the base function f (x) and a constant k, the function given by g (x) = f (x) k, can be sketched by shifting f (x) k units vertically Horizontal Translation For the base function f (x) and a constant k, the function given by g(x) = f (x k), can be sketched by shifting f (x) k units horizontally Vertical Stretches and ShrinksSimple and best practice solution for g(x)=k*f(x) equation Check how easy it is, and learn it for the future Our solution is simple, and easy to understand,

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