Slides Chapter 1. Measure Theory and Probability

1.8. RANDOM VARIABLES
Definition 1.49 The k-th moment of X with respect to the
mean µ is ∫
µ k := α k,µ = (x−µ) k dF X (x).
IR
In particular, second moment with respect to the mean is called
variance,
∫
σX 2 = V(X) = µ 2 = (x−µ) 2 dF X (x).
The standard deviation is σ X = √ V(X).
Definition 1.50 The k-th moment of X with respect to the
origin µ is
∫
α k := α k,0 = E(X n ) = x k dF X (x).
Proposition 1.20 It holds
µ k =
IR
k∑
( k
(−1) k−i i
i=0
IR
)
µ k−i α i .
Lemma 1.1 If α k = E(X k ) exists and is finite, then there exists
α m and is finite, ∀m ≤ k.
One way of obtaining information about the distribution of a
random variable is to calculate the probability of intervals of the
type (E(X) − ǫ,E(X) + ǫ). If we do not know the theoretical
distributionoftherandomvariablebutwedoknowitsexpectation
and variance, the Tchebychev’s inequality gives a lower bound of
this probability. This inequality is a straightforward consequence
of the following one.
ISABEL MOLINA 44