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Expected Values of Probability Functions

Module by: Don Johnson. E-mail the author

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The expected value of a function f· f · of a random variable XX is defined to be

EfX=∫-∞∞fxpXxdx f X x f x p X x (1)
Several important quantities are expected values, with specific forms for the function f· f · .
  • fX=X f X X . The expected value or mean of a random variable is the center-of-mass of the probability density function. We shall often denote the expected value by m X m X or just mm when the meaning is clear. Note that the expected value can be a number never assumed by the random variable ( pXm p X m can be zero). An important property of the expected value of a random variable is linearity: EaX=aEX a X a X , aa being a scalar.
  • fX=X2 f X X 2 . EX2 X 2 is known as the mean squared value of XX and represents the "power" in the random variable.
  • fX=X− m X 2 f X X m X 2 . The so-called second central difference of a random variable is its variance, usually denoted by σ X 2 σ X 2 . This expression for the variance simplifies to σ X 2 =EX2−EX2 σ X 2 X 2 X 2 , which expresses the variance operator σ·2 · . The square root of the variance σ X σ X is the standard deviation and measures the spread of the distribution of XX. Among all possible second differences X−c2 X c 2 , the minimum value occurs when c= m X c m X (simply evaluate the derivative with respect to cc and equate it to zero).
  • fX=Xn f X X n . EXn X n is the n th n th moment of the random variable and EX− m X n X m X n the n th n th central moment.
  • fX=ⅇⅈuX f X u X . The characteristic function of a random variable is essentially the Fourier Transform of the probability density function.
    EⅇⅈuX≡ Φ X ⅈu=∫-∞∞pXxⅇⅈuxdx u X Φ X u x p X x u x (2)
    The moments of a random variable can be calculated from the derivatives of the characteristic function evaluated at the origin.
    EXn=ⅈ-ndndun Φ X ⅈu|u,u=0 X n n u 0 u n n Φ X u (3)

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