Common Probability Theorems - Flashcard set - Statistics

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Central Limit Theorem

The Central Limit Theorem states that, for a large enough sample size, the distribution of the sample mean will be approximately normally distributed, regardless of the distribution of the population from which the samples are taken.

\text{If } X_i \text{ are iid, then } \quad \frac{1}{n}\sum_{i=1}^n X_i \to N(\mu, \frac{\sigma^2}{n})

Independent Events

Independent events are two or more events where the occurrence of one does not affect the occurrence of the other. It simplifies probability computations.

P(A \cap B) = P(A)P(B)

Variance of a Random Variable

Variance measures the spread of a set of numbers. In probability, it tells how much the outcomes of a random variable differ from the expected value.

\text{Var}(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2

Binomial Theorem

The Binomial Theorem describes the algebraic expansion of powers of a binomial. In probability, it's used to calculate the probabilities of success in binomial experiments.

P(X=k) = {n \choose k} p^k (1-p)^{n-k}

where

n

is the number of trials and

p

is the probability of success per trial.

Zero Probability

Zero Probability implies that an event is impossible within the context of a probability model. It's used to denote events that are outside the sample space or have no chance of occurring.

P(A) = 0

Multiplication Rule

The Multiplication Rule relates the probability of the intersection of two events to the probabilities of the individual events. It's used to find the probability that two events will occur in sequence.

P(A \cap B) = P(A)P(B|A)

Conditional Probability

Conditional Probability measures the probability of an event given that another event has occurred. It's a fundamental concept in probability for dealing with dependent events.

P(A|B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0

Law of Large Numbers

The Law of Large Numbers states that as the number of trials increases, the experimental probability of an event will get closer to the theoretical or expected probability. It underpins many statistical practices.

\lim_{n\to\infty} P_n(A) = P(A)

Complementary Probability

Complementary Probability refers to the probability that an event does not occur, complementary to the probability that it does. It's used to simplify calculations, especially for the probability of failure or non-occurrence.

P(A') = 1 - P(A)

Expected Value

Expected Value is the average outcome of a random variable over a large number of trials. It's used to determine the long-term average or expectation of a probability distribution.

E(X) = \sum_x xP(X=x)

Standard Deviation of a Random Variable

Standard Deviation is the square root of the variance. It's used in probability to measure the amount of variation or dispersion of a set of values.

\text{SD}(X) = \sqrt{\text{Var}(X)}

Chebyshev's Inequality

Chebyshev's Inequality gives a bound on the probability that the value of a random variable will be far from its mean. It's used to estimate the spread of a distribution when the exact distribution is unknown.

P(|X - \mu| \ge k\sigma) \le \frac{1}{k^2}

Law of Total Probability

The Law of Total Probability is a fundamental rule relating marginal probabilities to conditional probabilities. It partitions the sample space into exclusive and exhaustive segments. It's used in probability to calculate the probability of an event by considering all possible scenarios.

P(A) = \sum_{i} P(A|B_i)P(B_i)

Addition Rule

The Addition Rule is used to find the probability that either of two events will occur. It's important for events that are not mutually exclusive.

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Bayes' Theorem

Bayes' Theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. It's used in probability to update the probability estimate for an event as more evidence or information becomes available.

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

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