Probability & Distributions Classrooms
87.1 Introduction to Probability
Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
- Experiment: A process that leads to well-defined outcomes.
- Outcome: A single possible result of an experiment.
- Sample Space: The set of all possible outcomes of an experiment.
- Event: A subset of the sample space.
- Probability of an Event: $$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$
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87.2 Laws of Probability
Understanding the laws of probability is essential for calculating the likelihood of complex events. These laws govern how probabilities combine and interact.
- Addition Rule: For two events A and B:
- If mutually exclusive: $$P(A \text{ or } B) = P(A) + P(B)$$
- If not mutually exclusive: $$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$
- Multiplication Rule: For two events A and B:
- If independent: $$P(A \text{ and } B) = P(A) \times P(B)$$
- If dependent: $$P(A \text{ and } B) = P(A) \times P(B|A)$$ (where $$P(B|A)$$ is the probability of B given A)
- Complement Rule: $$P(A') = 1 - P(A)$$ (where $$A'$$ is the complement of A)
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87.3 Permutations & Combinations
Permutations and combinations are fundamental counting techniques used in probability to determine the number of ways events can occur.
- Permutation: An arrangement of objects in a specific order. Order matters. $$P(n, k) = \frac{n!}{(n-k)!}$$ where $n$ is the total number of items, and $k$ is the number of items to choose.
- Combination: A selection of objects where the order does not matter. $$C(n, k) = \frac{n!}{k!(n-k)!}$$ where $n$ is the total number of items, and $k$ is the number of items to choose.
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88.1 Binomial Distribution
The Binomial Distribution models the number of successes in a fixed number of independent Bernoulli trials.
- Conditions:
- Fixed number of trials ($n$).
- Each trial has only two possible outcomes (success/failure).
- Probability of success ($p$) is constant for each trial.
- Trials are independent.
- Probability Mass Function (PMF): $$P(X=k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k}$$ where $X$ is the number of successes, and $k$ is the specific number of successes.
- Mean: $$\mu = n \cdot p$$
- Variance: $$\sigma^2 = n \cdot p \cdot (1-p)$$
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88.2 Poisson Distribution
The Poisson Distribution models the number of events occurring in a fixed interval of time or space, given a constant average rate of occurrence.
- Conditions:
- Events occur independently.
- The rate of occurrence ($\lambda$) is constant.
- Two events cannot occur at exactly the same instant.
- Probability Mass Function (PMF): $$P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$$ where $X$ is the number of events, $k$ is the specific number of events, and $\lambda$ is the average rate of events.
- Mean: $$\mu = \lambda$$
- Variance: $$\sigma^2 = \lambda$$
🧪 Model Rare Events!
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89.1 Introduction to Normal Distribution
The Normal Distribution, also known as the Gaussian distribution or bell curve, is a continuous probability distribution that is symmetric about its mean. It is one of the most important distributions in statistics.
- Characteristics:
- Symmetric about the mean.
- Mean, median, and mode are equal.
- Tails approach the x-axis asymptotically.
- Area under the curve equals 1.
- Standard Normal Distribution: A special case with mean ($\mu$) = 0 and standard deviation ($\sigma$) = 1. $$Z = \frac{X - \mu}{\sigma}$$ where $Z$ is the Z-score, $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.
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89.2 Testing for Normality
Testing for normality helps determine if a dataset is well-modeled by a normal distribution. This is crucial for many statistical tests that assume normality.
- Graphical Methods:
- Histogram: Visually inspect for bell shape and symmetry.
- Normal Probability Plot (Q-Q Plot): Plots quantiles of the data against quantiles of a normal distribution. If data is normal, points fall along a straight line.
- Statistical Tests:
- Shapiro-Wilk Test: A widely used test for normality, especially for smaller sample sizes.
- Kolmogorov-Smirnov Test (Lilliefors Test): Another common test, particularly for larger sample sizes.
📊 Analyze Your Data's Distribution!
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