12 Expectations. Expectations 103

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1 Expectations Expectations At rst, for motivation purposes, the expectation of a r.v. will be introduced for the discrete case as the weighted mean value on the basis of the series value of an absolutely convergent series. The general expectation, basing on the integral, subsumes the special expectation for the discrete case. There are several properties of an expectation which allow a reasonably simple computational handling, cf Meaningful is the formation of the expectation with respect to the image measure, cf The expectation of a r.v. will be introduced as the weighted mean value on the basis of the series value of an absolutely convergent series. This special case is subsumed under the one of general probability spaces The expectation in the discrete case Let (Ω, P(Ω), P ) be a discrete probability space. Considering a real-valued r.v. X : Ω R, we are able

2 Expectations 104 to calculate not only the function value X(ω) for all ω Ω but also the probability P ({ω}). It is obvious that the average over these function values should be taken with the corresponding probabilities as weights; this provides a reason for the denition of the expectation as a nite sum or as the value of the series (12.1.1) X(ω)P ({ω}). ω Ω If Ω is nite, then (12.1.1) is a nite sum; if Ω is countable innite, then the value of the series (12.1.1), i.e. the expectation is dened to be the limes of the respective partial sums. If (12.1.1) is a nite sum or if the series given by (12.1.1) is absolutely convergent, then we say that X has the expectation E(X) := E P (X) with respect to P ; the expectation E P (X) is given by the value of the nite sum (12.1.1) or by the value of the series (12.1.1). The value of the series (12.1.1) exists, if the series (12.1.1) is absolutely convergent; in this case the

3 Expectations 105 value does not depend on the order of the summation. The countable basic space is not necessarily characterized by a natural order as in the case of N. E.g. think of Q.

4 Expectations Expectations with uncountable basic spaces Let (Ω, A, P ) be a probability space and X : (Ω, A, P ) (R, B) a r.v. (a measurable mapping). If the Integral (12.2.1) XdP of X with respect to the measure P exists and is nite, then we say, that X has the expectation E(X) := E P (X); the expectation E P (X) is given by (12.2.1) Special cases The denition (12.2.1) of the integral is so 'exible', that the denition 12.1 of the expectation in the discrete case is subsumed. This implies especially, that the rules for calculating with expectations are those of the integral (12.2.1), which are also valid for the expectation in the discrete case Important for applications is the probability space (R n, B n, P ) with the Borel σalgebra B n,

5 Expectations 107 n N, if the probability measure P has a Lebesguedensity, that means if P = f λ n. In this case we have ( ) XdP = X fdλ n = X f dx. ( ) is a Lebesgue integral, which for a great class of (integrand-) functions, (the so called regulated functions), can be evaluated by the respective Riemann integral. The following rules for expectations are rules for integrals and are especially a consequence of the fact, that expectations are integrals with respect to a probability measure, i.e. P (Ω) = 1 is essential Rules for expectations Let (Ω, A, P ) be an arbitrary (e.g. a discrete) probability space. Let X, Y be real valued r.v. and α, β R.

6 Expectations If X = 1 A, where 1 A is the indicator function (cf. Basics 1.3.1) for A A, then ( ) E P (X) = E P (1 A ) = P (A) If X(ω) = a R for all ω Ω, that means if X is the constant mapping with the the value a, then ( ) E P (X) = a If X and Y are integrable, which means that the expectations exists, then αx + βy is integrable and ( ) αe P (X) + βe P (Y ) = E P (αx + βy ) If X and Y are integrable, then (Linearity of the expectation) ( ) X(ω) Y (ω) (ω Ω) = E P (X) E P (Y ) If X is integrable, then (Isotony of the expectation) ( ) E(X) E( X ). The following fact the so called expectation with respect to the image measure is important, especially for modeling with r.v..

7 Expectations Theorem (Expectation with respect to the image measure) Let (Ω, A, P ) be a probability space, X : (Ω, A) (Ω, A ) be a r.v., P X be the distribution of X with respect to P and let h : (Ω, A ) (R, B) be a real r.v. Then: The real r.v. h(x(.)) has the expectation E P (h(x(.)) i h has the expectation E PX (h(.)) (with respect to the image measure P X ), and ( ) E P (h(x(.))) = E PX (h(.)) In the special case (Ω, A ) = (R, B) and h = id Ω we have (( )) E P (X(.)) = E PX ( id Ω (.)) As usual id Ω (.) is the identity mapping id Ω : Ω Ω with id Ω (ω ) = ω, (ω Ω ). The contents of ( ) is the subject of Experiment On the meaning of

8 Expectations 110 The statement , i.e. E P (X) = E PX ( id Ω ), means in particular, that the expectation E P (X) of X with respect to P is already determined by the distribution (image measure) P X. In fact, E P (X) can be determined without knowing neither X nor P. This is also expressed when speaking for short of the expectation of a distribution, for example the expectation of the binomial distribution B(n, p) as the expectation of a B(n, p)-distributed r.v.. The expectation is therefore a characteristic index of the distribution, i.e. the image measure Expectations of particular functions According to 12.6 the expectation of B(1, p) (of a B(1, p)- distributed r.v.) is (12.7.1) E B(1,p) ( id {0,1} ) = 0 (1 p) + 1 p = p. The expectations of the binomial distribution B(n, p), the Poisson distribution Π(λ) or of the normal distribution N(a, σ 2 ) can be determined in a similar way. One determines the expectation of id Ω with respect to

9 Expectations 111 the corresponding distribution B(n, p), Π(λ) or N(a, σ 2 ). (12.7.2) E B(n,p) ( id N 0 n ) = n p (12.7.3) E Π(λ) ( id N ) = λ (12.7.4) E N(a,σ 2 )( id R ) = a. (We renounce here consciously to present calculation techniques!) The notion of r.v. was introduced in as a measurable mapping. A special sight of the term random variable becomes meaningful when modeling stochastic phenomena The modeling of stochastic randomness with random variables We want to determine the expectation when throwing an unbiased dice. The interest thereby lies less on the result than on the modeling which reveals a special sight of the random variable. The number of eyes as the outcome of the dice-experiment

10 Expectations 112 is understood as the function value of a random variable X : (Ω, A, P ) (N 6, P(N 6 ), G 6 ), where G 6 is the discrete uniform distribution over N 6. The determination of the expectation of X stands here for a typical question of the dice-experiment, which requires to know the r.v. X and the probability measure P at least according to the denition of the expectation. According to ( ), E P (X) equals E PX ( id N 0 0 ). Therefore, E P (X) can be determined without knowledge of X and P, if only P X is known. When dening X to be the r.v. X : (Ω, A, P ) (N 6, P(N 6 ), G 6 ), as well X as also (Ω, A, P ) have here only a formal meaning. Knowing the distribution (image measure) P X, we need not know neither X nor (Ω, A, P), such that the intuitive meaning of a r.v. is often reduced to the knowledge of the distribution of the r.v., which is G 6 in the present example.

11 Expectations 113 To form the expectation one calculates now E P (X) = E PX ( id N6 ) = 6 i=1 i 1 6 = i=1 i = 7 2. For the expectation of a product of two independent r.v. holds a special product rule Theorem Let (Ω, A, P ) be a probability space and let X, Y : (Ω, A, P ) (R, B) be to stochastically independent r.v. with the expectations E P (X) and E P (Y ). Then (12.9.1) E P (X Y ) = E P (X) E P (Y ). The stochastic independence of X and Y entails (12.9.1); the validity of (12.9.1) however does not imply the stochastic independence of X and Y as can be shown with counter examples. (12.9.1) will be used again in the context of the covariance of two r.v.