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# Desigualdad de Cauchy-Schwarz

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## Sugeith Santos

on 9 December 2012

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#### Transcript of Desigualdad de Cauchy-Schwarz

Desigualdad de
Cauchy-Schwarz A demostrar: Definimos Regresando al caso h(t)=0 Sea X y Y variables con segundos
momentos finitos, entonces

para una constante c h(t)= E[(tX-Y) ]
desarrollando el cuadrado
=E[t X -2tXY+Y ]
=E[t X ]-E[2tXY]+E[Y ]
=t E[X ]-2tE[XY]+E[Y ]
=>
h(t) = at +bt+c
h(t)≥0 Yanela Paz Elizalde 142617
Ana Pía Cuenya Medrano 141353
Juan Ma. Murguía Pérez 141442
Sugeith Santos Domínguez 141449 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Casos Caso 1: h(t)=0

Si h(t)=0 => tiene 1 raíz real
E[(t X-Y) ]=0

h(t)=t E[X ]-2tE[XY]+E[Y ]=0 0 2 2 2 2 2 (E[XY]) =|E[XY]| ≤ E[X ]E[Y ] <=> P [Y=cX]=1 2 2 2 Fórmula Cuadrática 2E[XY]± √4E [XY]-4E[X ]E[Y ] t = 2E[X ] _________________________________ 0 2 2 2 2 ______________ 4E [XY]-4E[X ]E[Y ]=0

E [XY]=E[X ]E[Y ] 2 2 2 2 2 2 Caso 2: h(t)>0

h(t)= E[(tX-Y) ]>0

El discriminante es menor a 0

4E [XY]-4E[X ]E[Y ]<0
4E [XY]<4E[X ]E[Y ]
E [XY]<E[X ]E[Y ] 2 2 2 2 2 2 2 2 2 2 h(t)= E[(tX-Y) ]=0

E[W ]=0

V[W]=E[W ]-E [W]

E[W]=0
V[W]=0 sustituyendo sabemos que =>P[W=0]=1 Corolario Casos |ρ | ≤ 1 <=> una variable aleatoria es una función lineal dependiente de otra con probabilidad 1. X,Y Reescribimos la desigualdad de Cauchy-Schwarz como |E[UV]|≤ √E[U ]E[V ] y definimos U=X-µ V=Y-µ 2 2 x y Sustituimos |E[UV]| ≤ E[U ]E[V ] |E[UV]| E[U ]E[V ] ≤1 __________ |E[(X-µ )(Y-µ )]| E[(X-µ ) ]E[(Y-µ ) ] ___________________ ≤1 x y x y 2 2 2 2 2 2 |ρ | ≤ 1 X,Y Gracias Aplausos 2 2 2 ______________________
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