Probability Distributions

Probability Distributions

Discrete Probability Distribution:

• A function that assigns a probability to each possible value of a discrete random variable
• Must satisfy two conditions:
  1. The probability of each value is between 0 and 1
  2. The sum of the probabilities of all values is 1

Continuous Probability Distribution:

• A function that describes the probability of a continuous random variable taking on a given value
• Examples: Uniform Distribution, Normal Distribution, Exponential Distribution

Conditional Probability

Definition:

• The probability of an event occurring given that another event has occurred
• Notation: P(A|B) = Probability of event A occurring given that event B has occurred

Formula:

• P(A|B) = P(A ∩ B) / P(B)
• P(A ∩ B) = Probability of both events A and B occurring
• P(B) = Probability of event B occurring

Properties:

• P(A|B) ≥ 0
• P(A|B) ≤ 1
• P(A|B) + P(A'|B) = 1 (where A' is the complement of A)

Independent Events

Definition:

• Two events are independent if the occurrence of one event does not affect the probability of the other event
• Notation: P(A ∩ B) = P(A) × P(B)

Properties:

• P(A|B) = P(A) (since the occurrence of B does not affect the probability of A)
• P(B|A) = P(B) (since the occurrence of A does not affect the probability of B)

Random Variables

Definition:

• A variable whose possible values are determined by chance
• Can be discrete or continuous

Types of Random Variables:

• Discrete Random Variable: Takes on a countable number of distinct values
• Continuous Random Variable: Takes on an uncountable number of values in a given interval

Properties:

• Expected Value (Mean): The long-run average value of a random variable
• Variance: A measure of the spread or dispersion of a random variable
• Standard Deviation: The square root of the variance

Probability Distributions

• A discrete probability distribution assigns a probability to each possible value of a discrete random variable, satisfying two conditions: probabilities are between 0 and 1, and their sum is 1.
• A continuous probability distribution describes the probability of a continuous random variable taking on a given value, with examples including Uniform, Normal, and Exponential Distributions.

Conditional Probability

• Conditional probability is the probability of an event occurring given that another event has occurred, denoted as P(A|B).
• Formula: P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.
• Properties: P(A|B) ≥ 0, P(A|B) ≤ 1, and P(A|B) + P(A'|B) = 1, where A' is the complement of A.

Independent Events

• Two events are independent if the occurrence of one event does not affect the probability of the other event, denoted as P(A ∩ B) = P(A) × P(B).
• Properties: P(A|B) = P(A), and P(B|A) = P(B), since the occurrence of one event does not affect the probability of the other.

Random Variables

• A random variable is a variable whose possible values are determined by chance, and can be discrete or continuous.
• Types of random variables: discrete random variables take on a countable number of distinct values, while continuous random variables take on an uncountable number of values in a given interval.
• Properties of random variables: expected value (mean) is the long-run average value, variance measures the spread or dispersion, and standard deviation is the square root of the variance.

Sample Spaces

• A sample space is the set of all possible outcomes of an experiment.
• It is denoted by S or Ω (capital omega).
• The sample space for tossing a coin is {H, T} (heads or tails).

Independent Events

• Independent events are events where the occurrence of one event does not affect the probability of the other event.
• The probability of both events occurring is the product of their individual probabilities.
• The formula for independent events is P(A ∩ B) = P(A) × P(B).
• The probability of getting heads on both coins when tossing two coins is 0.5 × 0.5 = 0.25.

Probability Distributions

• A probability distribution is a function that describes the probability of each possible value of a random variable.
• Probability distributions can be discrete or continuous.
• Discrete uniform distribution is a type of distribution where each outcome has an equal probability.
• Binomial distribution models the number of successes in a fixed number of independent trials.
• Normal distribution (Gaussian distribution) is a continuous distribution with a symmetric bell-shaped curve.

Conditional Probability

• Conditional probability is the probability of an event occurring given that another event has occurred.
• The formula for conditional probability is P(A|B) = P(A ∩ B) / P(B).
• The probability of drawing a king given that the first card is a king is P(King|King) = P(King ∩ King) / P(King) = 3/4.
• This is calculated by dividing the probability of drawing two kings (P(King ∩ King) = 3/51) by the probability of drawing a king (P(King) = 4/52).