Working with Continuous Random Variables

Transcription

1 Working with Continuous Random Variables Friday, February 06, :00 PM Homework 1 posted, due Friday, February 27. The probability density function variable has the following meaning: of a random The probability density function is larger where the random variable is more likely to be near, but it is not a probability. (In particular, there's no reason that.) In fact for absolutely continuous random variables, by a direct consequence of the definition, for any. There are generalizations to "hybrid" random variables that are mostly continuously distributed but have some positive probability to take certain values, i.e., a Brownian particle could bind to the boundary and get stuck at the boundary. The PDF for (at some specified time t) could look like: AppSDE15 Page 1

2 The PDF for (at some specified time t) could look like: This really isn't a PDF in the standard sense of being a function, and in this example is not an absolutely continuous random variable. Really it's a hybrid random variable which has a discrete part and an absolutely continuous part. Alternatively, one can compute correctly by simply generalizing the notion of PDFs to allow generalized functions involving Dirac delta functions for the discrete components. So since we said a PDF doesn't actually tell you the probability directly (without integrating over some set), what is the meaning of the probability density? Let's consider the case where calculate: is continuous. Then, we can when is small, by mean value theorem. AppSDE15 Page 2

3 So when is continuous at a point, then: That is, it describes the density of the probability at in a parallel way that mass density describes the density of mass near some spatial location. Everything above carries over without change to cases where the state space is finite dimensional. and the probability density has the intuitive meaning that: where d is the dimension of the state space S. Note that probabilities have no dimensions, but probability density has dimensions of where is the dimension of one coordinate of the state space. In one dimension, we have the simple relationship between PDF and CDF: As we've seen, the multidimensional generalization of the PDF is not problematic, but the multidimensional generalization of the CDF is possible but very awkward. Properties of Random Variables AppSDE15 Page 3

4 In complex models, one has some input random variables that are explicitly modeled, say and the variables of interest will generally be some functions/functionals of these specified random variables, so the outputs of a model are often of the form where is some relationship between input and output variables, sometimes explicit, sometimes implicit. It can be exceedingly difficult to develop theoretical expressions or even numerical approximations for the full probability distribution of in a complex system. Attempting to do this is an active research area called "uncertainty quantification" (UQ). So in practice, both theoretically and computationally, the point is not try to specify the full probability distribution of the output random variables, but just some key properties (or "statistics") of these random variables. Let's called the output random variables. The most fundamental property of a random variable expected value or average or expectation: is its mean or The second most fundamental property of a random variable is the size of its uncertainty, which is characterized by its variance and standard deviation. The variance measures the size of the noisy second term in terms of its mean square. Let's do this for now just for scalar random variables (one dimension): AppSDE15 Page 4

5 More intuitive measure of the noisiness of the random variable is via the standard deviation: which is the root-mean-square size of the fluctuations of the random variable about its mean. It has a size corresponding to typical deviations of realizations of the random variable from its mean. For vector random variables, one can view each component as a onedimensional random variable and measure its variance. There is actually a more general concept of "second moments" of a vector random variable that gives you the covariance matrix of the vector; we'll talk about that later. Higher order moments give finer information about the random variable, but the return on investment decreases rapidly after the second moment: third moments skewness (asymmetry about mean) fourth moments kurtosis (fatness of tails relative to Gaussian model) Note the following identity for variance: Also note well the law of the unconscious statistician (LUS) which gives you a very useful alternative for computing the basic statistics of a function of a random variable: So this is crucial to being able to do both analytical and numerical calculations of properties/statistics of the output random variable without having to calculate the full probability distribution of the output random variable. That is, one should specify what properties of the random variable are desired, and then do the calculation accordingly. Some Fundamental Scalar Random Variable Models AppSDE15 Page 5

6 Some Fundamental Scalar Random Variable Models Uniform distribution : parameters a < b Exponential distribution : one parameter AppSDE15 Page 6

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