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A compilation of properties of the fundamental concept of mathematical expectation. Not all of these properties are used explicitly in this treatment, but they are included for reference.
E [ g ( X ) ] = g ( X ) d P

We suppose, without repeated assertion, that the random variables and Borel functions of random variables or random vectors are integrable. Useof an expression such as I M ( X ) involves the tacit assumption that M is a Borel set on the codomain of X .

  • E [ a I A ] = a P ( A ) , any constant a , any event A
  • E [ I M ( X ) ] = P ( X M ) and E [ I M ( X ) I N ( Y ) ] = P ( X M , Y N ) for any Borel sets M , N (Extends to any finite product of such indicator functions of random vectors)
  • Linearity . For any constants a , b , E [ a X + b Y ] = a E [ X ] + b E [ Y ] (Extends to any finite linear combination)
  • Positivity; monotonicity .
    1. X 0 a . s . implies E [ X ] 0 , with equality iff X = 0 a . s .
    2. X Y a . s . implies E [ X ] E [ Y ] , with equality iff X = Y a . s .
  • Fundamental lemma . If X 0 is bounded, and { X n : 1 n } is a.s. nonnegative, nondecreasing, with lim n X n ( ω ) X ( ω ) for a.e. ω , then lim n E [ X n ] E [ X ]
  • Monotone convergence . If for all n , 0 X n X n + 1 a . s . and X n X a . s . ,
    then E [ X n ] E [ X ] (The theorem also holds if E [ X ] = )

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  • Uniqueness . * is to be read as one of the symbols , = , or
    1. E [ I M ( X ) g ( X ) ] * E [ I M ( X ) h ( X ) ] for all M iff g ( X ) * h ( X ) a . s .
    2. E [ I M ( X ) I N ( Z ) g ( X , Z ) ] = E [ I M ( X ) I N ( Z ) h ( X , Z ) ] for all M , N iff g ( X , Z ) = h ( X , Z ) a . s .
  • Fatou's lemma . If X n 0 a . s . , for all n , then E [ lim inf X n ] lim inf E [ X n ]
  • Dominated convergence . If real or complex X n X a . s . , | X n | Y a . s . for all n , and Y is integrable, then lim n E [ X n ] = E [ X ]
  • Countable additivity and countable sums .
    1. If X is integrable over E , and E = i = 1 E i (disjoint union), then E [ I E X ] = i = 1 E [ I E i X ]
    2. If n = 1 E [ | X n | ] < , then n = 1 | X n | < a . s . and E [ n = 1 X n ] = n = 1 E [ X n ]
  • Some integrability conditions
    1. X is integrable iff both X + and X - are integrable iff | X | is integrable.
    2. X is integrable iff E [ I { | X | > a } | X | ] 0 as a
    3. If X is integrable, then X is a.s. finite
    4. If E [ X ] exists and P ( A ) = 0 , then E [ I A X ] = 0
  • Triangle inequality . For integrable X , real or complex, | E [ X ] | E [ | X | ]
  • Mean-value theorem . If a X b a . s . on A , then a P ( A ) E [ I A X ] b P ( A )
  • For nonnegative, Borel g , E [ g ( X ) ] a P ( g ( X ) a )
  • Markov's inequality . If g 0 and nondecreasing for t 0 and a 0 , then
    g ( a ) P ( | X | a ) E [ g ( | X | ) ]
  • Jensen's inequality . If g is convex on an interval which contains the range of random variable X , then g ( E [ X ] ) E [ g ( X ) ]
  • Schwarz' inequality . For X , Y real or complex, | E [ X Y ] | 2 E [ | X | 2 ] E [ | Y | 2 ] , with equality iff there is a constant c such that X = c Y a . s .
  • Hölder's inequality . For 1 p , q , with 1 p + 1 q = 1 , and X , Y real or complex,
    E [ | X Y | ] E [ | X | p ] 1 / p E [ | Y | q ] 1 / q
  • Minkowski's inequality . For 1 < p and X , Y real or complex,
    E [ | X + Y | p ] 1 / p E [ | X | p ] 1 / p + E [ | Y | p ] 1 / p
  • Independence and expectation . The following conditions are equivalent.
    1. The pair { X , Y } is independent
    2. E [ I M ( X ) I N ( Y ) ] = E [ I M ( X ) ] E [ I N ( Y ) ] for all Borel M , N
    3. E [ g ( X ) h ( Y ) ] = E [ g ( X ) ] E [ h ( Y ) ] for all Borel g , h such that g ( X ) , h ( Y ) are integrable.
  • Special case of the Radon-Nikodym theorem If g ( Y ) is integrable and X is a random vector, then there exists a real-valued Borel function e ( · ) , defined on the range of X , unique a.s. [ P X ] , such that E [ I M ( X ) g ( Y ) ] = E [ I M ( X ) e ( X ) ] for all Borel sets M on the codomain of X .
  • Some special forms of expectation
    1. Suppose F is nondecreasing, right-continuous on [ 0 , ) , with F ( 0 - ) = 0 . Let F * ( t ) = F ( t - 0 ) . Consider X 0 with E [ F ( X ) ] < . Then,
      ( 1 ) E [ F ( X ) ] = 0 P ( X t ) F ( d t ) and ( 2 ) E [ F * ( X ) ] = 0 P ( X > t ) F ( d t )
    2. If X is integrable, then E [ X ] = - [ u ( t ) - F X ( t ) ] d t
    3. If X , Y are integrable, then E [ X - Y ] = - [ F Y ( t ) - F X ( t ) ] d t
    4. If X 0 is integrable, then
      n = 0 P ( X n + 1 ) E [ X ] n = 0 P ( X n ) N k = 0 P ( X k N ) , for all N 1
    5. If integrable X 0 is integer-valued, then E [ X ] = n = 1 P ( X n ) = n = 0 P ( X > n ) E [ X 2 ] = n = 1 ( 2 n - 1 ) P ( X n ) = n = 0 ( 2 n + 1 ) P ( X > n )
    6. If Q is the quantile function for F X , then E [ g ( X ) ] = 0 1 g [ Q ( u ) ] d u

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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