Expected Values of Probability Functions 1.4 2003/05/13 2003/08/08 12:31:12.457 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Kevin Duh kevinduh@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu The expected value of a function f · of a random variable X is defined to be f X x f x p X x Several important quantities are expected values, with specific forms for the function f · . 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 or just m when the meaning is clear. Note that the expected value can be a number never assumed by the random variable ( p X m can be zero). An important property of the expected value of a random variable is linearity: a X a X , a being a scalar. f X X 2 . X 2 is known as the mean squared value of X and represents the "power" in the random variable. 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 . This expression for the variance simplifies to σ X 2 X 2 X 2 , which expresses the variance operator · . The square root of the variance σ X is the standard deviation and measures the spread of the distribution of X. Among all possible second differences X c 2 , the minimum value occurs when c m X (simply evaluate the derivative with respect to c and equate it to zero). f X X n . X n is the n th moment of the random variable and X m X n the n th central moment. f X u X . The characteristic function of a random variable is essentially the Fourier Transform of the probability density function. u X Φ X u x p X x u x The moments of a random variable can be calculated from the derivatives of the characteristic function evaluated at the origin. X n n u 0 u n n Φ X u