2.4 Marginal and conditional distributions

Marginal and conditional distributions are key concepts in probability theory. They help us understand how random variables interact and influence each other. These tools allow us to analyze complex systems by breaking them down into simpler components.

By mastering these concepts, we can tackle real-world problems in various fields. From making informed decisions to predicting outcomes, marginal and conditional distributions provide a solid foundation for working with uncertainty and probability.

Joint probability distributions

• Fundamental concept in probability theory describes the probability of two or more random variables occurring simultaneously
• Provides a complete description of the relationship between multiple random variables
• Can be represented as a probability mass function (PMF) for discrete random variables or a probability density function (PDF) for continuous random variables

Marginal distributions

• Probability distribution of a single random variable derived from a joint distribution by summing or integrating out the other variables
• Represents the unconditional probability of a random variable without considering the values of other variables

Marginal PMF

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• Obtained by summing the joint PMF over all possible values of the other random variables
• For discrete random variables $X$ and $Y$, the marginal PMF of $X$ is given by $P_X(x) = \sum_y P_{X,Y}(x,y)$
• Example: If $X$ and $Y$ represent the outcomes of two dice rolls, the marginal PMF of $X$ is the probability of obtaining a specific value on the first die, regardless of the value on the second die

Marginal PDF

• Obtained by integrating the joint PDF over all possible values of the other random variables
• For continuous random variables $X$ and $Y$, the marginal PDF of $X$ is given by $[f_X(x)](https://www.fiveableKeyTerm:f_x(x)) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) dy$
• Example: If $X$ and $Y$ represent the heights of a randomly selected male and female, the marginal PDF of $X$ is the probability density of the male height, regardless of the female height

Obtaining marginal distributions

• Marginal distributions can be derived from the joint distribution by summing (discrete case) or integrating (continuous case) over the other variables
• Allows for the analysis of individual random variables without considering the effects of other variables
• Useful in simplifying complex joint distributions and understanding the behavior of specific random variables

Conditional distributions

• Probability distribution of a random variable given the value of another random variable
• Describes the probability of an event occurring given that another event has already occurred

Conditional PMF

• For discrete random variables $X$ and $Y$, the conditional PMF of $X$ given $Y=y$ is defined as $P_{X|Y}(x|y) = \frac{P_{X,Y}(x,y)}{P_Y(y)}$
• Represents the probability of $X$ taking a specific value given that $Y$ has already taken a specific value
• Example: If $X$ and $Y$ represent the outcomes of two dice rolls, the conditional PMF of $X$ given $Y=6$ is the probability of obtaining a specific value on the first die, given that the second die shows a 6

Conditional PDF

• For continuous random variables $X$ and $Y$, the conditional PDF of $X$ given $Y=y$ is defined as $f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}$
• Represents the probability density of $X$ taking a specific value given that $Y$ has already taken a specific value
• Example: If $X$ and $Y$ represent the heights of a randomly selected male and female, the conditional PDF of $X$ given $Y=170$ cm is the probability density of the male height, given that the female height is 170 cm

Calculating conditional distributions

• Conditional distributions can be calculated using the definition of conditional probability
• Involves dividing the joint probability by the marginal probability of the conditioning variable
• Allows for the analysis of the relationship between random variables and how the value of one variable affects the probability distribution of another

Independence of random variables

• Two random variables are independent if their joint probability distribution is equal to the product of their marginal distributions
• Independence implies that the occurrence of one event does not affect the probability of the other event

Definition of independence

• For discrete random variables $X$ and $Y$, independence is defined as $P_{X,Y}(x,y) = P_X(x) \cdot P_Y(y)$ for all $x$ and $y$
• For continuous random variables $X$ and $Y$, independence is defined as $f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)$ for all $x$ and $y$
• Example: If $X$ and $Y$ represent the outcomes of two fair dice rolls, they are independent because the probability of obtaining any specific combination of values is equal to the product of the probabilities of obtaining each value individually

Properties of independent variables

• If $X$ and $Y$ are independent, then $E[XY] = E[X] \cdot E[Y]$ (the expected value of the product is equal to the product of the expected values)
• If $X$ and $Y$ are independent, then $Var(X+Y) = Var(X) + Var(Y)$ (the variance of the sum is equal to the sum of the variances)
• Independence simplifies many probability calculations and is a crucial assumption in various statistical models and techniques

Relationship between joint, marginal, and conditional distributions

• Joint, marginal, and conditional distributions are interconnected and can be derived from one another using probability rules

Product rule for discrete variables

• The joint PMF can be expressed as the product of the marginal PMF and the conditional PMF: $P_{X,Y}(x,y) = P_X(x) \cdot P_{Y|X}(y|x) = P_Y(y) \cdot P_{X|Y}(x|y)$
• Allows for the calculation of the joint distribution from the marginal and conditional distributions
• Example: If $X$ represents the type of car (sedan or SUV) and $Y$ represents the color (red, blue, or green), the joint PMF can be calculated by multiplying the marginal PMF of car type and the conditional PMF of color given the car type

Product rule for continuous variables

• The joint PDF can be expressed as the product of the marginal PDF and the conditional PDF: $f_{X,Y}(x,y) = f_X(x) \cdot f_{Y|X}(y|x) = f_Y(y) \cdot f_{X|Y}(x|y)$
• Allows for the calculation of the joint distribution from the marginal and conditional distributions
• Example: If $X$ and $Y$ represent the heights of a randomly selected male and female, the joint PDF can be calculated by multiplying the marginal PDF of male height and the conditional PDF of female height given the male height

Applications of marginal and conditional distributions

• Marginal and conditional distributions have numerous applications in various fields, including statistics, machine learning, and decision-making

Bayesian inference

• Marginal and conditional distributions play a crucial role in Bayesian inference, which involves updating the probability of a hypothesis based on new evidence
• The prior probability (marginal) is combined with the likelihood (conditional) to obtain the posterior probability using Bayes' theorem: $P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}$
• Example: In a medical diagnosis, the prior probability of a disease (based on population data) is updated with the likelihood of observing specific symptoms given the presence of the disease to obtain the posterior probability of the disease given the observed symptoms

Decision making under uncertainty

• Marginal and conditional distributions are used in decision-making problems where the outcomes are uncertain
• The expected value of a decision can be calculated using the marginal distribution of the random variable representing the outcome and the conditional distribution of the payoff given the outcome
• Example: In a business investment decision, the marginal distribution of market conditions (recession or growth) and the conditional distribution of profits given the market conditions can be used to calculate the expected value of the investment and make an informed decision