M4L4 Expectation and Moments of Functions of Random Variable 1. Introduction Functions of random variable are discussed in previous lectures. In this lecture, properties of random variable, e.g. expectation, moments and moment-generating function of ‘functions of random variable’ are discussed in detail. 2. Usefulness of the properties of random variables The properties of random variables are useful for statistical problems in civil engineering. Standard procedure can be followed to obtain the moments and expectations from the probability distribution functions, as described in previous lectures. There are different methods to obtain pdf of the functions of random variable available, though the procedure might be complex in few cases. For such instances, information of moments of the derived random variables is very useful. 3. Expectation of functions of discrete random variables Expectation of the discrete random variable X is given by, Therefore, the expectation of the function of a discrete random variable, is expressed as, Expectation of the continuous random variable X is given by, 3.1. Properties of Expectation The following properties of expectation hold good. a. , if ‘ ’ is constant. b. , if ‘ ’ is constant. c. for two constants ‘ ’ and ‘ ’. d. The expectation operator and functions of random variables do not commute, i.e. 3.2. Example for discrete random variables Problem 1. The random variable has a probability mass function (pmf) and . Find the mean of the function for . Solution. The mean of the function, 3.3. Example for discrete random variables Problem 2. Assume that the inter-arrival time, of a vehicle approaching a toll station of a bridge has an exponential pdf with parameter . There are Thus toll lines in that toll station. vehicles can be accommodated at a time. Determine the mean arrival time of vehicles and the coefficient of variation of this arrival time. Assume that the arrivals are independent of each other (Kottegoda and Rosso, 2008). Solution. As the inter-arrival time, variance of are and The total time for the arrival of where, follows an exponential distribution, the mean and the respectively. vehicle is denoted as . Thus we get, is the arrival time of the th vehicle and Hence the coefficient of variation is: Thus the variation of arrival time decreases with increase in toll lines, as ‘ ’ is a positive real number. 4. Moments of functions of random variables In general, the And the th th moment of the function of discrete random variable moment of the function of continuous random variable is given by: is given by: 4.1. Moments of functions of random variables about its mean The th moment of the function of discrete random variable th moment of the functions of continuous functions about its mean is given by: The is expressed as: 4.2. Variance of discrete functions The variance of discrete function, is expressed as: 4.3. Variance of continuous functions The variance of discrete function, is expressed as: 5. Mean and Variance of Linear Function Let us consider a linear function as, , where a and b are constants. The mean values of is mathematical expectation of , i.e. Similarly, variance of can be expressed as, 6. Expansion of Functions of Random Variable The function of random variable, value, can be expanded in a Taylor series about the mean . 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 length of two rods will be determined by two measurements with an unbiased instruments with an unbiased instrument that make random error with mean and standard deviation in each measurement. Compute the variance in the estimation of the length and by the following methods: a. The two rods are measured separately b. The sum and difference of the length of two rods are measured instead of individual lengths (Ang and Tang, 1975) Solution. a. Let and denote the measurement obtained for the two rods, then where, are the errors involved in the measurements. The variance in the estimation of is: Similarly, b. Suppose denotes that the measured combined length of the two rods and denotes the measured difference between the lengths of the two rods. Then, Solving these two equations, we get, and Assuming that the errors are statically independent, the variance in the estimation of is therefore, Similarly, From this, we can further find that the second method of measuring the length of the rods is better, since the variance in the estimation of the true lengths and , are smaller.