M4L18 - M4L5 Expectation and Moments of Functions of Random...

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    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 about its mean is given by: The th moment of the functions of continuous functions is expressed as: 2.2. Variance of discrete functions The variance of discrete function, is expressed as:  
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    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
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M4L18 - M4L5 Expectation and Moments of Functions of Random...

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