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Chapter 4 Distributions Lectures 13 - 17 In this chapter, we associate with random variable, a nondecreasing function called distribution function and study its properties. Definition 4.1. (Distribution function) Let . The function be defined by is called the distribution function of Theorem 4.0.13 The distribution function has the following properties (i) (ii) is non decreasing. (iii) is right continuous. Proof: (i) Let , set Then Therefore i.e., Hence Using similar argument, we can prove (ii) For . Hence (iii) We have to show that, for each Let . Set . a random variable on Then and Therefore Theorem 4.0.14 Proof. Let The set discontinuity points of a distribution function be the distribution function of a random variable is countable. and Then where Set Then Also note that for If be distinct points in . Then Therefore Hence is countable. Theorem 4.0.15 Let define be a random variable on a probability space . For , Then is a probability measure on . Proof. Also for all . Let [note that be pairwise disjoint, then 's are disjoint imply that 's are disjoint] This completes the proof. Definition 4.2 The probability measure and we denote it by Note that if is called the distribution or law of the random variable . is the distribution function of , then Example 4.0.1 Consider the probability space Define as follows. For = number of heads in Then takes values from given by , . . The distribution function of is The distribution function of the random variable is given by Example 4.0.2 (Bernoulli distribution) On Then is a probability measure on , for , define called the Bernoulli distribution. A random variable with Bernoulli distribution is called a Bernoulli random variable. For example: Let and define the probability measure Define on such that by Then Hence is a Bernoulli random variable. The distribution function corresponding to Bernoulli distribution is given by Example 4.0.3 (Binomial distribution with parameters given by ).On ,the probability measure Example 4.0.29 Example 4.0.30 Example 4.0.31 Example 4.0.32 Classification of Random Variables. Random variables can be classified using Distribution function Defination 4.3: Defination 4.4: Defination 4.5: Defination 4.5: A continuous random variable with pdf is said to be an absolutely continuous random variable. Example 4.0.33: