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Probability Theory Fundamentals
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Sample Space
The set of all possible outcomes of a probabilistic experiment. Example: For a coin toss, the sample space is {Heads, Tails}.
Event
A subset of the sample space to which a probability is assigned. Example: Getting a Head in a coin toss is an event.
Probability of an Event
A measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1. Example: Probability of getting Heads in a coin toss is 0.5.
Independent Events
Two events are independent if the occurrence of one does not affect the probability of the other. Example: Tossing two different coins.
Conditional Probability
The probability of an event given that another event has occurred. Given by , if . Example: Probability of drawing an ace from a deck of cards, given that a heart has been drawn.
Random Variable
A variable whose possible values are numerical outcomes of a random phenomenon. Example: The sum of the roll of two dice.
Probability Mass Function (PMF)
A function that gives the probability that a discrete random variable is exactly equal to some value. Example: PMF of a fair dice roll giving the probability of each number 1 through 6 is 1/6.
Probability Density Function (PDF)
A function that describes the relative likelihood for a continuous random variable to take on a given value. Example: The normal distribution curve is a PDF.
Cumulative Distribution Function (CDF)
A function that gives the probability that a random variable is less than or equal to a certain value. Example: CDF of a standard normal distribution at z=0 is 0.5.
Expected Value
The theoretical average of a random variable over many trials. Calculated as for a discrete variable. Example: Expected value of a dice roll is 3.5.
Variance
A measure of the spread of a distribution about the expected value. Given by . Example: Variance of a dice roll is about 2.92.
Standard Deviation
The square root of the variance, a measure of the spread of a distribution. Example: Standard deviation of a dice roll is about 1.71.
Covariance
A measure of the extent to which two random variables change together. Given by . Example: Covariance between height and weight.
Correlation Coefficient
A normalized measure of the strength and direction of the relationship between two random variables. Given by . Example: Correlation between study time and exam score.
Law of Large Numbers
States that as the number of trials increases, the sample average will converge to the expected value. Example: Flipping a coin many times will get close to an average of 0.5 heads.
Central Limit Theorem
States that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution. Example: The average height of randomly selected people will be normally distributed if the sample size is large.
Bayes' Theorem
A formula that describes how to update the probabilities of hypotheses when given evidence. Given by . Example: Revising the probability of having a disease, given a positive test result.
Discrete Uniform Distribution
A probability distribution where each of the n finite outcomes has equal probability 1/n. Example: Rolling a fair six-sided dice.
Poisson Distribution
A discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. Given by for Example: Number of emails received in an hour.
Binomial Distribution
A probability distribution that summarizes the likelihood that a value will take one of two independent states across a number of observations or trials. Given by for . Example: Number of heads in 10 coin tosses.
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