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Conditional PMFs, given the value of another random variable Properties of conditional PMFs. Contents Conditioning (1) - conditioning a random variable on an event - conditioning one random variable on another - summary of facts about conditional PMFs Conditioning (2) - conditional expectation - total expectation theorem + mean and variance of the geometric (example 2.17.) + The Two-Envelopes Pa..

Conditional PMFs, given the value of another random variable Properties of conditional PMFs. Contents Conditioning (1) - conditioning a random variable on an event - conditioning one random variable on another - summary of facts about conditional PMFs Conditioning (2) - summary of facts about conditional expectation - total expectation theorem + mean and variance of the geometric (example 2.17.)..

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..