In this lecture we discuss how to derandomize algorithms. We will see a brute force algorithm (enumeration) for derandomization. We will also see that some random algorithms do not need true randomness. Specifically, we will see an example where only pairwise random bits are needed. Next, we will see how we can generate pairwise random values and how this conservation on the amount of randomnes...

Similarly we can get the distribution function of Y easily from the joint distribution function of X and Y : FY (y) = lim x→∞ (x, y) = F (∞, y). The distribution functions FX and FY are sometimes called the marginal distribution functions of X and Y respectively. The joint distribution function F of X and Y contains all the statistical information about X and Y . In particular, given the joint ...

This paper introduces the concept of a random variable, which is nothing more than a variable whose numeric value is determined by the outcome of an experiment. To describe the probabilities that are associated with these numeric values in a concise and conceptually useful manner, the probability distribution and probability density function are introduced. Then, the moment generating function ...

If Equation (1) be satisfied, then all sets S are equally likely under the null hypothesis. To obtain a permutation test that is both unbiased and most powerful, one need only select the set S so that its probability under the alternative >0 is a maximum. This is accomplished in most cases by rejecting the hypothesis for all values of the test statistic that lie in the upper tail of the permuta...

The aim of this paper is to establish a theory of random variables on domains. Domain theory is a fundamental component of theoretical computer science, providing mathematical models of computational processes. Random variables are the mainstay of probability theory. Since computational models increasingly involve probabilistic aspects, it’s only natural to explore the relationship between thes...

A large chunk of probability is about random variables. Instead of giving a precise definition, let us just mention that a random variable can be thought of as an uncertain, numerical (i.e., with values in R) quantity. While it is true that we do not know with certainty what value a random variable X will take, we usually know how to assign a number the probability that its value will be in som...

Math 394 1 (Almost bullet-proof) Definition of Expectation Assume we have a sample space Ω, with a σ−algebra of subsets F , and a probability P , satisfying our axioms. Define a random variable as a a function X : Ω → R, such that all subsets of Ω of the form {ω |a < X(ω) ≤ b}, for any real a ≤ b are events (belong to F). Assume at first that the range of X is bounded, say it is contained in th...

With more than one random variable, the set of outcomes is an N -dimensional space, Sx = {−∞ < x1, x2, · · · , xN < ∞}. For example, describing the location and velocity of a gas particle requires six coordinates. • The joint PDF p(x), is the probability density of an outcome in a volume element dx = ∏N i=1 dxi around the point x = {x1, x2, · · · , xN}. The joint PDF is normalized such that px(...

1 Probabilities In talking about probabilities, the fundamental object is Ω, the sample space. Points (elements) in Ω are denoted (generically) by ω. We assign probabilities to subsets of Ω. Assume for the moment that Ω is finite or countably infinite. Thus Ω could be the space of all possible outcomes when a coin is tossed three times in a row or say, the set of positive integers. A probabilit...