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Lecture 8: Continuous Random Variables MIT opencourse 강좌를 보고 내용을 나름대로 정리한 글입니다. 각 개념이 궁금하실 때마다 해당 부분을 보고 개념을 잡으시는 것에 도움이 되면 좋겠습니다. Chapter 2까지는 영어로 책 기반으로 블로그 글을 작성 중이었는데, 정보 정리와 전달에 있어 효율적이지 않은 것 같다고 판단했습니다. 그래서 Chapter 3부터는 강의를 기준으로 개괄적이고 중요한 내용들을 저의 insight를 담아 적어보려 합니다. 그럼 시작합니다! 목차 Continuous random variables (r.v.) and probability density functions (PDF) Means and variances Cumulative dist..

- independence related to random variables. - suitable events involving the possible values of various random variables Independence of a Random Variable from an Event 'The occurrence of the conditioning event provides no new information on the value of the random variable.' The random variable \(X\) is independent of the event \(A\) if $$P(X = x and A) = P(X = x)P(A) = p_X(x)P(A), \quad \text{f..

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..

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 is independent of B We have introduced the conditional probability \(P(A | B)\) to capture the partial information that event B provides about event A. The occurrence of B provides no such information and does not alter the probability that A has occurred, i.e. , \(P(A | B) = P(A)\). Note that by the definition \(P(A | B) = P(A \cap B)/P(B)\), this is equivalent to \(P(A \cap B) = P(A)P(B)\). ..

Total Probability Theorem Let \(A_1, \cdots , \ A_n\) be disjoint events that form a partition of the sample space (each possible outcome is included in exactly one of the events A1, ... , An) and assume that \(P(A_i) > 0\), for all \(i\). Then, for any event B, we have $$P(B) = P(A_1 \cap B) + ... + P(A_n \cap B) = P(A_1)P(B I A_1) + ... + P(A_n )P(B I A_n).$$ The theorem is visualized and prov..