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Questions and Answers
What does the notation P(A|B) represent?
What does the notation P(A|B) represent?
What is the formula for conditional probability?
What is the formula for conditional probability?
What is the property of conditional probability that states P(A|B) 0?
What is the property of conditional probability that states P(A|B) 0?
What is the definition of a discrete random variable?
What is the definition of a discrete random variable?
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What is the term for the set of all possible values of a random variable?
What is the term for the set of all possible values of a random variable?
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What is the function that describes the probability of each possible value of a discrete random variable?
What is the function that describes the probability of each possible value of a discrete random variable?
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What is the average of the squared differences between a random variable and its expectation?
What is the average of the squared differences between a random variable and its expectation?
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What is the theorem that states P(A|B) = P(B|A) * P(A) / P(B)?
What is the theorem that states P(A|B) = P(B|A) * P(A) / P(B)?
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What is the term for events A and B being conditionally independent given C?
What is the term for events A and B being conditionally independent given C?
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Study Notes
Conditional Probability
- Definition: The probability of an event occurring given that another event has occurred.
- Notation: P(A|B) reads "the probability of A given B"
- Formula: P(A|B) = P(A ∩ B) / P(B)
- Properties:
- P(A|B) ≥ 0 (non-negativity)
- P(A|B) ≤ 1 (boundedness)
- P(A|B) + P(A'|B) = 1 (completeness)
- P(A ∩ B) = P(A|B) * P(B) (multiplication rule)
- Conditional Independence: Events A and B are conditionally independent given C if P(A|B,C) = P(A|C)
- Bayes' Theorem: P(A|B) = P(B|A) * P(A) / P(B)
Random Variables
- Definition: A variable whose possible values are determined by chance
- Types:
- Discrete Random Variables (DRVs): take on a countable number of distinct values
- Continuous Random Variables (CRVs): take on any value within a certain range or interval
- Probability Distribution: a function that describes the probability of each possible value of a random variable
- Notation: X for a random variable, x for a specific value of X
- Terminology:
- Support: the set of all possible values of a random variable
- Probability Mass Function (PMF): function describing the probability of each value of a DRV
- Probability Density Function (PDF): function describing the probability of each value of a CRV
- Cumulative Distribution Function (CDF): function describing the probability that a random variable takes on a value less than or equal to x
- Expectation: the long-run average value of a random variable
- Variance: the average of the squared differences between a random variable and its expectation
Conditional Probability
- The probability of an event A occurring given that event B has occurred is represented as P(A|B)
- The formula to calculate conditional probability is P(A|B) = P(A ∩ B) / P(B)
- Conditional probability has four properties: non-negativity, boundedness, completeness, and multiplication rule
- Two events A and B are conditionally independent given C if P(A|B,C) = P(A|C)
- Bayes' Theorem is P(A|B) = P(B|A) * P(A) / P(B)
Random Variables
- A random variable is a variable whose possible values are determined by chance
- There are two types of random variables: Discrete Random Variables (DRVs) and Continuous Random Variables (CRVs)
- A probability distribution is a function that describes the probability of each possible value of a random variable
- The notation X represents a random variable, and x represents a specific value of X
Random Variable Notation and Terminology
- The support of a random variable is the set of all possible values it can take
- A Probability Mass Function (PMF) is a function that describes the probability of each value of a DRV
- A Probability Density Function (PDF) is a function that describes the probability of each value of a CRV
- A Cumulative Distribution Function (CDF) is a function that describes the probability that a random variable takes on a value less than or equal to x
Expectation and Variance
- The expectation of a random variable is its long-run average value
- The variance of a random variable is the average of the squared differences between the variable and its expectation
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Description
This quiz covers the concept of conditional probability, including its definition, notation, formula, and properties. It also touches on conditional independence and Bayes' Theorem.