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:
    1. Fixed number of trials ($n$).
    2. Each trial has only two possible outcomes (success/failure).
    3. Probability of success ($p$) is constant for each trial.
    4. 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:
    1. Events occur independently.
    2. The rate of occurrence ($\lambda$) is constant.
    3. 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: $$\sigma^2 = \lambda$$

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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:
    1. Symmetric about the mean.
    2. Mean, median, and mode are equal.
    3. Tails approach the x-axis asymptotically.
    4. 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.

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