Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
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<strong>1.</strong>8. RANDOM VARIABLES<br />
Definition <strong>1.</strong>49 The k-th moment of X with respect to the<br />
mean µ is ∫<br />
µ k := α k,µ = (x−µ) k dF X (x).<br />
IR<br />
In particular, second moment with respect to the mean is called<br />
variance,<br />
∫<br />
σX 2 = V(X) = µ 2 = (x−µ) 2 dF X (x).<br />
The st<strong>and</strong>ard deviation is σ X = √ V(X).<br />
Definition <strong>1.</strong>50 The k-th moment of X with respect to the<br />
origin µ is<br />
∫<br />
α k := α k,0 = E(X n ) = x k dF X (x).<br />
Proposition <strong>1.</strong>20 It holds<br />
µ k =<br />
IR<br />
k∑<br />
( k<br />
(−1) k−i i<br />
i=0<br />
IR<br />
)<br />
µ k−i α i .<br />
Lemma <strong>1.</strong>1 If α k = E(X k ) exists <strong>and</strong> is finite, then there exists<br />
α m <strong>and</strong> is finite, ∀m ≤ k.<br />
One way of obtaining information about the distribution of a<br />
r<strong>and</strong>om variable is to calculate the probability of intervals of the<br />
type (E(X) − ǫ,E(X) + ǫ). If we do not know the theoretical<br />
distributionofther<strong>and</strong>omvariablebutwedoknowitsexpectation<br />
<strong>and</strong> variance, the Tchebychev’s inequality gives a lower bound of<br />
this probability. This inequality is a straightforward consequence<br />
of the following one.<br />
ISABEL MOLINA 44