Stat 5101 Lecture Notes - School of Statistics

2.4. MOMENTS 37The Multiplicativity Non-PropertyOne might suppose that there is a property analogous to the additivityproperty, except with multiplication instead of additionE(XY )=E(X)E(Y), Uncorrelated X and Y only! (2.8)As the editorial comment says, this property does not hold in general. We willlater see that when (2.8) does hold we have a special name for this situation:we say the variables X and Y are uncorrelated.Taking a Function Outside an ExpectationSuppose g is a linear function defined bywhere a and b are real numbers. Theng(x) =a+bx, x ∈ R, (2.9)E{g(X)} = g(E{X}), Linear g only! (2.10)is just Theorem 2.2 stated in different notation. The reason for the editorialcomment is that (2.10) does not hold for general functions g, only for linearfunctions. Sometime you will be tempted to use (2.10) for a nonlinear functiong. Don’t! Remember that it is a “non-property.”For example, you may be asked to calculate E(1/X) for some random variableX. The “non-property,” if it were true, would allow to take the functionoutside the expectation and the answer would be 1/E(X), but it isn’t true, and,in general( ) 1E ≠ 1X E(X)There may be a way to do the problem, but the “non-property” isn’t it.2.4 MomentsIf k is a positive integer, then the real numberis called the k-th moment of the random variable X.If p is a positive real number, then the real numberα k = E(X k ) (2.11)β p = E(|X| p ) (2.12)is called the p-th absolute moment of the random variable X.If k is a positive integer and µ = E(X), then the real numberµ k = E{(X − µ) k } (2.13)