M4L5. Expectation and Moments of Functions of Random Variable

Transcription

1 M4L5 Expectation and Moments of Functions of Random Variable 1. Introduction This lecture is a continuation of previous lecture, elaborating expectations, moments and moment generating functions of the functions of random variable discussed earlier. 2. Moments of functions of random variables In general, the th moment of the function of discrete random variable is given by: And the th moment of the function of continuous random variable is given by: 2.1. Moments of functions of random variables about its mean The th moment of the function of discrete random variable by: The th moment of the functions of continuous functions about its mean is given is expressed as: 2.2. Variance of discrete functions The variance of discrete function, is expressed as:

2 2.3. Variance of continuous functions The variance of discrete function, is expressed as: 3. Mean and Variance of Linear Function Let us consider a linear function as,, where and are constants. The mean values of is mathematical expectation of, i.e. Similarly, variance of can be expressed as, Problem 1. The random variable has a probability mass function (pmf) for and. Find the variance of the function. Solution. The mean of the function, Problem 2. Consider a simple case where the variable can only takes a value of or. This situation can represent the occurrence of a flood at a particular site on a river, where the event is the exceedence of a specified flow in the river. Let the probability of such an occurrence be. The event is the complementary event and has a probability of occurrence of. The probabilities of the two events are given by the Bernoulli distribution. for. Find the variance. (Kottegoda and Rosso, 2008) Solution. If we take the second moment about the origin, then using the expression for variance,, we get:

3 Let, now as, the variance is given by, 4. Expansion of Functions of Random Variable The function of random variable, value,. can be expanded in a Taylor series about the mean Y 2 1 dg 1 2 d g g( ) ( ) ( ) 2 1! d 2! d where derivatives are evaluated at. If the series is truncated at linear terms, then the first-order approximate mean and variance of are obtained. The variance of function of random variable, It should be noted that, if the function, is approximately linear for the entire range of value, then above two equations will yield good approximation of exact moments. Problem 3. The maximum impact pressure, determined by: of ocean waves on coastal structures is where, =density of water; =length of hypothetical piston; =thickness of air cushion and =horizontal velocity of the advancing wave. Suppose that the mean crest velocity is ft/s with a coefficient of variation, of. The density of sea water is about slugs/cu.ft and the ratio. Determine the mean and standard deviation of the peak impact pressure. Solution. We have So, Similarly,

4 So, 5. Moment Generating Functions for Derived Random Variable A random variable that is a function of other random variables and its probability distribution are also defined as a derived variable and a derived variable, respectively. The determination of the probability distribution of a derived variable from those of the basic variables is a difficult task. So, the evaluation of its moment generating functions can provide some useful information on the target variable Definition of Moment Generating Function The moment generating function of a random variable is defined as. If the moment generating function (mgf) exists, its th derivative at the origin is the th order central moment of. If is a random variable taking integer values, then by definition, its moment generating function is: Similarly if is a random variable taking continuous values, the mgf is: Notes The basic concept is that if two random variables have identical moment generating functions, then they possess the same probability distribution. The procedure is to find the moment generating function and then compare it to all known ones to see if there is a match. This is most commonly done to see if a distribution approaches the normal distribution as the sample size tends to infinity Theorem 1 Let Then, F x and y F Y are two cumulative distribution functions whose moments exist. i. If and have bounded support, then F u and u for all integers F Y are equal for all if and only if

5 ii. If the moment generating functions exist and for all t in some neighborhood of, then F u F u Y for all. 5.4 Theorem 2 Differentiating the equation of mgf for times, we obtain: 5.3. Corollary If, then If are independent and, then Problem 4. Let and, Find, given Solution. Here we get, So, that means has a normal distribution with mean and variance. Problem 5. The impact pressure of sea waves on coastal structures may be evaluated as, where is the horizontal velocity of the advancing wave and is a constant. Because of the uncertainty involved in the evaluation of, we consider this to be a random variable; is is thus a derived variable from. Assume has mean and standard deviation and has the normal pdf. Solution. Let. with zero mean and unit variance and pdf Now,

6 Representing and by substituting in moment generating equation: So, Here to note that, using the transformation curve is unity. and also the area under the Taking the first derivative of the mgf at the origin, we can obtain the mean of as, Similarly, taking the second derivative, second-order moment of W is obtained as: Hence, the variance of W is: The mean of the required impact pressure expectation as, follows immediately from the linear property of Thus the variance is represents the coefficient of variation of horizontal velocity of the advancing wave, then the mean of can be obtained as: which equals to augmented by a factor The variance of, which equals to augmented by a factor of.