15. Marginal Distributions
Marginal distributions are vital in understanding individual variables within multivariable distributions. They are created by integrating or summing over other variables, enabling focus on specific probabilities in various applications, especially in engineering fields. The chapter presents the necessary mathematical foundations and practical implications of marginal distributions, emphasizing their importance in multivariate analysis.
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What we have learnt
- Marginal distributions provide insights into individual variables in multivariate distributions.
- They are derived by integrating (or summing) the joint distribution over the other variables.
- In engineering, marginal distributions are used in probabilistic modeling and signal analysis.
- Understanding marginals is key to simplifying complex systems and focusing on specific variables of interest.
Key Concepts
- -- Joint Probability Distribution
- A function that gives the probability of two continuous random variables occurring together.
- -- Marginal Distribution
- The probability distribution of a single variable irrespective of others, obtained by integrating the joint distribution.
- -- Marginalization
- The process of removing one or more variables by integrating their effects out.
- -- Independence
- A condition where the joint distribution of variables equals the product of their marginal distributions.
- -- Probability Density Function (pdf)
- A function that describes the likelihood of a continuous random variable to take on a particular value.
- -- Probability Mass Function (pmf)
- A function that gives the probability of discrete random variables taking specific values.
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