5.1 Joint PMFs and PDFs for multiple random variables

Joint PMFs and PDFs are key tools for understanding how multiple random variables interact. They give us the probability of specific outcomes occurring together, like rolling two dice or measuring someone's height and weight.

These functions help us calculate probabilities for complex events involving multiple variables. We can use them to find marginal and conditional distributions, check for independence, and compute important statistics like expected values and covariances.

Joint Probability Mass Functions

Definition and Notation

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• A joint probability mass function (PMF) is a function that gives the probability that a discrete random vector $(X, Y)$ takes on a specific value $(x, y)$
• The joint PMF is denoted as $P(X = x, Y = y)$ or $[p_{X,Y}(x, y)](https://www.fiveableKeyTerm:p_{x,y}(x,_y))$, where $X$ and $Y$ are discrete random variables and $x$ and $y$ are specific values they can take
• Example: Consider rolling two fair six-sided dice. The joint PMF would give the probability of obtaining specific pairs of values (1, 1), (1, 2), etc.

Properties and Domain

• The joint PMF satisfies two properties:
  • Non-negativity: $P(X = x, Y = y) \geq 0$ for all $x$ and $y$
  • The sum of probabilities over all possible values of $(x, y)$ equals 1
• The domain of a joint PMF is the set of all possible pairs $(x, y)$ that the random vector $(X, Y)$ can take
• The joint PMF can be represented in tabular form or as a formula, depending on the nature of the random variables and their relationship
• Example: For the two dice example, the domain would be all pairs $(x, y)$ where $x, y \in \{1, 2, 3, 4, 5, 6\}$

Joint Probability Density Functions

Definition and Notation

• A joint probability density function (PDF) is a function that describes the probability distribution of a continuous random vector $(X, Y)$
• The joint PDF is denoted as $[f_{X,Y}(x, y)](https://www.fiveableKeyTerm:f_{x,y}(x,_y))$, where $X$ and $Y$ are continuous random variables and $x$ and $y$ are specific values they can take
• Example: The joint PDF of the height and weight of adult males in a population would describe the probability distribution of these two continuous variables

Properties and Domain

• The joint PDF satisfies two properties:
  • Non-negativity: $f_{X,Y}(x, y) \geq 0$ for all $x$ and $y$
  • The double integral of the joint PDF over the entire domain equals 1
• The probability of $(X, Y)$ falling within a specific region $A$ is given by the double integral of the joint PDF over that region: $P((X, Y) \in A) = \iint_A f_{X,Y}(x, y) \, dx \, dy$
• The domain of a joint PDF is the set of all possible pairs $(x, y)$ that the random vector $(X, Y)$ can take, which is typically a continuous region in the $xy$-plane
• Example: The domain for the height and weight joint PDF would be all possible pairs of positive real numbers

Probabilities from Joint Distributions

Discrete Random Variables

• For discrete random variables, the probability $P(X = x, Y = y)$ is directly given by the joint PMF evaluated at $(x, y)$
• To find the probability of a specific event involving discrete random variables, sum the joint PMF values for all $(x, y)$ pairs that satisfy the event's conditions
• Example: In the dice example, to find $P(X + Y = 7)$, sum the probabilities of all pairs $(x, y)$ where $x + y = 7$

Continuous Random Variables

• For continuous random variables, the probability $P((X, Y) \in A)$ is given by the double integral of the joint PDF over the region $A$
• To find the probability of a specific event involving continuous random variables, evaluate the double integral of the joint PDF over the region that satisfies the event's conditions
• When working with joint PDFs, it is essential to identify the limits of integration based on the given event or region
• Example: To find the probability that the height is between 170 and 180 cm and the weight is between 70 and 80 kg, integrate the joint PDF over this rectangular region

Properties of Joint Distributions

Marginal and Conditional Distributions

• The joint probability distribution (PMF or PDF) provides a complete description of the probabilistic behavior of a random vector $(X, Y)$
• The marginal PMF or PDF of a single random variable (e.g., $X$) can be obtained by summing (for discrete) or integrating (for continuous) the joint PMF or PDF over all possible values of the other variable (e.g., $Y$)
• The conditional PMF or PDF of one random variable given a specific value of the other variable can be calculated using the joint and marginal PMFs or PDFs
• Example: In the height and weight example, the marginal PDF of height can be found by integrating the joint PDF over all possible weights

Independence and Expected Values

• Independence of random variables can be assessed using the joint PMF or PDF. If $P(X = x, Y = y) = P(X = x)P(Y = y)$ for all $x$ and $y$ (discrete case) or $f_{X,Y}(x, y) = f_X(x)f_Y(y)$ for all $x$ and $y$ (continuous case), then $X$ and $Y$ are independent
• The expected value, variance, and covariance of random variables can be calculated using the joint PMF or PDF, providing insights into the relationship between the variables and their individual behaviors
• Example: The covariance between height and weight can be calculated using the joint PDF to measure the linear relationship between the two variables