MA2216/ST2131 Probability Notes 1 A Brief Historical Account. Three hundred and fifty years ago, it is said, correspondences between two famous French mathematicians, Blaise Pascal and Pierre de Fermat, gave birth to a mathematical theory of probability.
MA2216/ST2131 Probability Notes 10 Properties of Expectation and Inequalities 1. Summary of Basic Properties. Let us first review some elementary properties of mathematical expectation, and then develop and exploit additional properties and useful calcula
MA2216/ST2131 Probability Notes 9 Review & Examples and Properties of Expectation 1. Linear Transformation. Let X1 , X2 , . . . , Xn be continuous random variables having joint density f and let random variables Y1 , Y2 , . . . , Yn be defined by the foll
MA2216/ST2131 Probability Notes 8 Multidimensional Changes of Variables and Bivariate Normal Distribution 1. Changes of Variables Let X1 and X2 be jointly distributed random variables with joint probability density function fX1 ,X2 . It is sometimes neces
MA2216/ST2131 Probability Notes 7 Sums of Independent Random Variables Very often we are interested in the sum of independent random variables. When X and Y are independent, we would like to know the distribution of X + Y . In the following, we will deal
MA2216/ST2131 Probability Notes 6 Joint Probability Distributions So far, we have only considered probability distributions for single random variables. There will be situations, however, where we may find it desirable to record the simultaneous outcomes
MA2216/ST2131 Probability Notes 5 Distribution of a Function of a Random Variable We have seen that an "affine transformation" of a normal r.v. is still normally distributed. To be precise, let X N (, 2 ). Put Y = aX + b. Then, it has been pointed out in
MA2216/ST2131 Probability Notes 4 1. Discrete Uniform Distribution. 1. If X assumes the values x1 < x2 < . < xn , with equal probabilities, then X is called a discrete uniform r.v. The probability distribution of X is called the discrete uniform distribut
MA2216/ST2131 Probability Notes 3 1. Random Variables. It is frequently the case that, when an experiment is performed, we are mainly interested in some function of the outcome as opposed to the actual outcome itself. For instance, in testing 100 electron
MA2216/ST2131 Probability Notes 2 1. Axioms of Probability Consider an experiment whose sample space is S. The objective of probability is to assign to each event A a number IP(A), in [0, 1], called the probability of the event A, which will give a precis
MA2216/ST2131 Probability Notes 11 Central Limit Theorem Before proceeding to the central limit theorem, we will first review the topics such as moment generating functions as well as conditional expectation, and then consider a few examples for reference