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Section 5 (cont.): Moments of a Random Variable, Moment Generating Function October 7th, 2014 Lesson 10 In an earlier lecture, we defined the n-th moment of the random variable X to be the number E [X n ]. The variance Var [X ] = E [X 2 ] − (E [X ])2 is in fact equal to E [(X − µX )2 ], the 2nd central moment of X about the mean µX . In this lecture we will introduce the idea of the moment generating function of a random variable X . Lesson 10 The moment generating function (mgf) of a random variable X is denoted by MX (t) (sometimes M(t), mX (t), or m(t)), and it is defined as MX (t) = E [e tX ]. If X is a discrete random variable, then MX (t) = P tx e p(x ). If X is a continuous random variable, then MX (t) = R∞ −∞ e tx f (x ) dx . Lesson 10 Here are some useful facts about moment generating functions. It is always the case that MX (0) = 1. We can find the moments of X by taking the derivatives of MX (t): MX0 (0) = E [X ], MX00 (0) = E [X 2 ], (n) MX (0) = E [X n ] We also have d dt ln[MX (t)] = t=0 MX0 (0) MX (0) = E [X ] and d2 dt 2 ln[MX (t)] t=0 = Var [X ] Lesson 10 Example (5.8) The pdf of X is f (x ) = 5e −5x for x > 0. Find the moment generating function of X and use it to find the first and second moments of X , and the variance of X . Lesson 10 Example (5.9) The moment generating function of X is given as t < α, where α > 0. Find Var [X ]. Lesson 10 α α−t for Example (5.10) If the moment generating function for the random variable X 1 is MX (t) = 1−t , find the third moment of X about the point x = 3. Lesson 10 Example (5.11) Let X1 , X2 , X3 be a random sample from a discrete distribution with probability function 1/4 p(x ) = 3/4 0 for x = 0 for x = 1 otherwise Determine the moment generating function, M(t), of Y = X1 X2 X3 . Lesson 10 Here we define some concepts that are in the section but will not come up in the exercises. For a random variable X , the skewness of X is defined to be E [(X − µ)3 ]/σ 3 . If skewness is positive, we say the distribution is skewed to the right. If it is negative, we say the distribution is skewed to the left. 2 d 00 If h is a function such that dx 2 h(x ) = h (x ) ≥ 0 for all points x with non-zero density or probability for X , then E [h(X )] ≥ h(E [X ]). The inequality reverses if h00 ≤ 0. This is known as Jensen’s inequality. Lesson 10 For a finite collection of random variables X1 , . . . , Xk with density or probability functions f1 (x ), . . . , fk (x ), and for P real numbers 0 ≤ αi ≤ 1 with ki=1 αi = 1 (these are called “weights”), then we can construct a new random variable X with density function f (x ) = α1 f1 (x ) + · · · + αk fk (x ). Then for X we have E [X n ] = α1 E [X1n ] + · · · + αk E [Xkn ] MX (t) = α1 MX1 (t) + · · · + αk MXk (t) Lesson 10