Important Facts about Random Variables Friday, February 01, 2013 2:01 PM Homework 1 hopefully posted over the weekend. Two more fundamental concepts about random variables. Often it is useful to summarize a random variable not in complete detail (as would be obtained from a PDF or CDF) but in terms of numbers. The most important summary descriptor of a random variable Y is the mean or expectation. The second most important statistic of a random variable is its variance: The meaning of variance is more transparent in terms of its square root, which is called the Standard deviation This is the root-mean-square deviation of the random variable about its mean. This describes, in the most basic way, the variability of the random variable. AppSDE13 Page 1
Examples of Random Variables Uniform distribution The discontinuity of the PDF plays no role because anything involving the PDF involves integration and discontinuities are harmless when integrated. Exponential distribution AppSDE13 Page 2
If one wanted to go beyond mean and standard deviation to encode, i.e., asymmetries of the probability distribution about the mean, then one could go to a third order moment, and develop a skewness from that. And one can go to arbitrarily high order moments, but doing this only makes sense in the context of particular questions and applications -- the first two moments (mean and variance) are by far the most fundamental. Moments: Actually cumulants are a more efficient way to represent higher order information relative to moments, i.e., removing redundancies. Simplest example is that the variance is the second order cumulant. Normal (Gaussian) distribution AppSDE13 Page 3
Let's start thinking about dealing with a family of random variables. The simplest setting is when these random variables are independent. Intuitively, this means the values that one of the random variables take is completely unrelated to the values that the other random variables will take. AppSDE13 Page 4
A collection of random variables Are said to be identically distributed when each random variable in the collection has the same probability distribution. In this case, we simply say they all have a common probability distribution: Or in terms of probability density Central Limit Theorem Under some mild conditions (PDFs of the random variables should decay fast enough -- Lindeberg and Lyapunov conditions) sums of many independent, identically distributed random variables can be well approximated by a Gaussian distribution. In fact, this result can be generalized to allow weak correlation between the random variables, but the conditions for this are technical (mixing, etc.). Also the random variables don't need to be quite identically distributed -- so long as their sizes are not too disparate. Simplest Model for Brownian Motion Single Brownian particle, observed at discrete moments of time. AppSDE13 Page 5
Model: What properties should the random variables Z n have? Mean zero because the solvent is isotropic. Identically distributed: time homogeneity. Independence: This relies on separation of time scales. On the one hand, the water molecules themselves have rather short mean free paths, meaning they change direction fairly rapidly And surely the time between experimental observations is long compared to this. And this would surely justify saying that the force the Brownian particle experiences during one experimental interval is independent of the next. But be careful -- the particle has inertia, and so its response (as manifested by the kick random variables) takes place over a longer time, and for micron size particles, this is So this implies that the changes in the motion to the Brownian particle are roughly independent over successive time intervals if the length of the time interval satisfies A priori, we don't have a basis for assuming the probability distribution for the random kicks Z n. But consider the following argument: Choose a time scale AppSDE13 Page 6
We could have formulated a simple model just as above, but with the time interval between successive "observations" as All the above assumptions apply to this model (Z n is mean zero and iid). And therefore by the central limit theorem, we see that Z n should be well approximated by a Gaussian distribution. AppSDE13 Page 7