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Probability Distributions

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Probability distributions are key in statistics, outlining how probabilities are assigned to outcomes of random processes. They include discrete distributions, represented by probability mass functions for countable outcomes, and continuous distributions, depicted by probability density functions for uncountable outcomes. The text delves into cumulative distribution functions, uniform distributions, and the binomial probability distribution, highlighting their importance in quantifying uncertainty and modeling random events across various fields.

Exploring Probability Distributions

Probability distributions are fundamental concepts in statistics that describe how probabilities are assigned to different outcomes of a random process. The sample space, comprising all possible outcomes, is central to defining a probability distribution. These distributions can be depicted through probability mass functions for discrete variables or probability density functions for continuous variables. For example, the probability distribution of rolling a fair six-sided die can be represented by a probability mass function where each outcome from 1 to 6 has an equal probability of \( \frac{1}{6} \).
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Discrete and Continuous Probability Distributions

Probability distributions are broadly classified into discrete and continuous types. Discrete probability distributions, or probability mass functions, are applicable when the set of possible outcomes is finite or countable, such as the number of heads resulting from flipping a coin multiple times. These distributions require that the probabilities of all outcomes are non-negative and sum to 1. Continuous probability distributions, or probability density functions, are used for uncountably infinite outcomes, like measuring the height of people. Here, probabilities are not assigned to individual outcomes but to intervals, and the area under the distribution curve represents the probability for a given range, with the total area under the curve being 1.

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00

Define Probability Distributions

Probability distributions assign probabilities to outcomes of a random process.

01

Difference: PMF vs PDF

PMF applies to discrete variables, PDF to continuous variables.

02

Example: Fair Die Roll Distribution

Each outcome (1-6) has an equal probability of 1/6.

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