introduction-probability.pdf

If the expected value exists then it computes to e f

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If the expected value exists, then it computes to E f = n =1 f ( ω n ) q n [ -∞ , ] . A simple example for the expectation is the expected value while rolling a die: Example 3.1.8 Assume that Ω := { 1 , 2 , . . . , 6 } , F := 2 Ω , and P ( { k } ) := 1 6 , which models rolling a die. If we define f ( k ) = k, i.e. f ( k ) := 6 i =1 i 1I { i } ( k ) , then f is a measurable step-function and it follows that E f = 6 i =1 i P ( { i } ) = 1 + 2 + · · · + 6 6 = 3 . 5 .
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3.2. BASIC PROPERTIES OF THE EXPECTED VALUE 49 Besides the expected value, the variance is often of interest. Definition 3.1.9 [variance] Let (Ω , F , P ) be a probability space and f : Ω R be an integrable random variable. Then var(f) = σ 2 f = E [f - E f] 2 [0 , ] is called variance . Let us summarize some simple properties: Proposition 3.1.10 (1) If f is integrable and α, c R , then var( α f - c) = α 2 var(f) . (2) If E f 2 < then var(f) = E f 2 - ( E f) 2 < . Proof . (1) follows from var( α f - c) = E [( α f - c) - E ( α f - c)] 2 = E [ α f - α E f] 2 = α 2 var(f) . (2) First we remark that E | f | ≤ ( E f 2 ) 1 2 as we shall see later by H¨ older’s in- equality (Corollary 3.6.6), that means any square integrable random variable is integrable. Then we simply get that var(f) = E [f - E f] 2 = E f 2 - 2 E (f E f) + ( E f) 2 = E f 2 - 2( E f) 2 + ( E f) 2 . 3.2 Basic properties of the expected value We say that a property P ( ω ), depending on ω , holds P -almost surely or almost surely (a.s.) if { ω Ω : P ( ω ) holds } belongs to F and is of measure one. Let us start with some first properties of the expected value. Proposition 3.2.1 Assume a probability space , F , P ) and random vari- ables f, g : Ω R . (1) If 0 f ( ω ) g ( ω ) , then 0 E f E g . (2) The random variable f is integrable if and only if | f | is integrable. In this case one has | E f | ≤ E | f | .
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50 CHAPTER 3. INTEGRATION (3) If f = 0 a.s., then E f = 0 . (4) If f 0 a.s. and E f = 0 , then f = 0 a.s. (5) If f = g a.s. and E f exists, then E g exists and E f = E g . Proof . (1) follows directly from the definition. Property (2) can be seen as follows: by definition, the random variable f is integrable if and only if E f + < and E f - < . Since ω Ω : f + ( ω ) = 0 ω Ω : f - ( ω ) = 0 = and since both sets are measurable, it follows that | f | = f + + f - is integrable if and only if f + and f - are integrable and that | E f | = | E f + - E f - | ≤ E f + + E f - = E | f | . (3) If f = 0 a.s., then f + = 0 a.s. and f - = 0 a.s., so that we can restrict ourselves to the case f ( ω ) 0. If g is a measurable step-function with g = n k =1 a k 1I A k , g ( ω ) 0, and g = 0 a.s., then a k = 0 implies P ( A k ) = 0. Hence E f = sup { E g : 0 g f, g is a measurable step-function } = 0 since 0 g f implies g = 0 a.s. Properties (4) and (5) are exercises. The next lemma is useful later on. In this lemma we use, as an approximation for f , the so-called staircase -function. This idea was already exploited in the proof of Proposition 2.1.3. Lemma 3.2.2 Let , F , P ) be a probability space and f : Ω R be a random variable.
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