exercise:E5b7173007: Difference between revisions
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Let <math>X,Y</math> be any two random variables. | Let <math>X,Y</math> be any two random variables with a joint density function. Suppose that <math display = "block">\operatorname{E}[X|Y] = g(Y), \, g(y) = E[X|Y=y].</math> | ||
< | Which of the following statements is always true: | ||
<ul class="mw-excansopts"> | |||
<li><math>|\operatorname{Cov}(X,Y)| < |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|</math></li> | <li><math>|\operatorname{Cov}(X,Y)| < |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|</math></li> | ||
<li><math>|\operatorname{Cov}(X,Y)| > |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|</math></li> | <li><math>|\operatorname{Cov}(X,Y)| > |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|</math></li> | ||
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<li>If <math>\operatorname{Cov}(\operatorname{E}[X | Y],Y) = 0</math> then <math>X </math> and <math>Y</math> are independent.</li> | <li>If <math>\operatorname{Cov}(\operatorname{E}[X | Y],Y) = 0</math> then <math>X </math> and <math>Y</math> are independent.</li> | ||
<li>If <math>\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)</math> for every <math>Y</math> then <math>X = \operatorname{E}[X | Y]</math>.</li> | <li>If <math>\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)</math> for every <math>Y</math> then <math>X = \operatorname{E}[X | Y]</math>.</li> | ||
</ | </ul> | ||
Latest revision as of 22:00, 19 July 2024
Let [math]X,Y[/math] be any two random variables with a joint density function. Suppose that
[[math]]\operatorname{E}[X|Y] = g(Y), \, g(y) = E[X|Y=y].[[/math]]
Which of the following statements is always true:
- [math]|\operatorname{Cov}(X,Y)| \lt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
- [math]|\operatorname{Cov}(X,Y)| \gt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
- [math]\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)[/math]
- If [math]\operatorname{Cov}(\operatorname{E}[X | Y],Y) = 0[/math] then [math]X [/math] and [math]Y[/math] are independent.
- If [math]\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)[/math] for every [math]Y[/math] then [math]X = \operatorname{E}[X | Y][/math].