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Given a random variable X, one may generate other random variables by applying various transformations on X. If Y = g(X) is a function of a random variable X, then Y is also a random variable, since it provides a numerical value for each possible outcome. This is because every outcome in the sample space defines a numerical value x for X and hence also the numerical value y = g(x) for Y. If X is..

The most important way to characterize a random variable is through the probabilities of the values that it can take. Notation of PMF. Probability Mass Function of X, denoted \(p_X\). $$p_X(x) = P(X = x) = P({\omega \in \Omega \text{ s.t. } X(\omega ) = x})$$ x is any possible value of X, the probability mass of x denoted \(p_X (x)\). \(p_X (x)\) is the probability of the event \(\{X = x\}\) con..

Connectivity with previous chapters To simply take the concepts from Chapter 1 (probabilities, conditioning, independence, etc.) and apply them to random variables rather than events, together with some convenient new notation. numerical values를 가지고 있으면 often useful to assign probabilities to them. Lecture note 기준 Random variables An assignment of a value (number) to every possible outcome Mathe..

A set is a collection of objects, which are the elements of the set. x is an element of S, $$x\in S$$ x is not an element of S, $$x\notin S$$ S have no elements, empty set : $$\emptyset$$ Finite, Countably infinite, uncountable If \(S\) contains a finite number of elements, \(S\) is finite; $$S = \{x_1, x_2, \cdots, x_n\}.$$ If \(S\) contains infinitely many elements, \(S\) is countably infinite..