Mixed Random Variables, Joint Probability Distribution

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Lecture 26 Agenda 1. Mixed random variables and the the importance of distribution function 2. Joint Probability Distribution for discrete random variables Mixed random variables and the the impor- tance of distribution function Any numerical quantity associated with a random experiment is called a random variable. Mathematically we say, any function X : S R is a random variable. We have already defined random variables like this in the beginning of the course, and then we considered a special class of random variables, namely the discrete random variables. We know that for a discrete random variable X , when somebody asks me about it’s probability distribution, I have to provide him two things 1. Range(X) = { x 1 , x 2 , x 3 , . . . } = The set of values that X takes. 2. and, for each x i Range ( X ), I have to provide P ( X = x i ). Then we learned how to calculated the expectation, variance etc for a discrete random variable and studied various examples like Binomial, Geo- metric, Negative Binomial ...... After that we considered continuous random variables, for which P ( X = x ) = 0 for all x R . We saw that if some one asked me about the distribution of a continuous random variable X , I have to give him the density f X : R [0 , ), such that 1. for all a < b , P ( a < X < b ) = R b a f X ( x ) dx 2. and, R -∞ f X ( x ) dx = 1. Then similar to discrete random variables, we learned how to calculate mean, variance etc for a continuous random variable. We also saw various examples of continuous random variables like exponential, gamma, beta etc. Now a natural question to ask would be —– Does every random vari- able has to be either discrete or continuous ? Is there no other kind of random variable ?
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