6.2 Joint probability density functions for continuous random variables

Joint probability density functions describe how two or more continuous random variables behave together. They're essential for understanding complex systems where multiple factors interact, like in weather patterns or financial markets.

These functions have key properties, including non-negativity and integration to 1. They can be visualized as 3D surfaces or contour plots, helping us grasp the likelihood of different combinations of values occurring simultaneously.

Joint Probability Density Functions

Properties of joint probability density

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• Joint probability density function (joint PDF) describes the probability distribution of two or more continuous random variables simultaneously
  • Notation: $f_{X,Y}(x,y)$ represents the joint PDF for random variables $X$ and $Y$
• Non-negative property: joint PDF values are always greater than or equal to zero ($f_{X,Y}(x,y) \geq 0$) for all possible values of $x$ and $y$
• Integration property: the double integral of the joint PDF over its entire domain equals 1 ($\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{X,Y}(x,y)dxdy = 1$)
  • Ensures the total probability of all possible outcomes is 100%
• Marginal PDFs obtained by integrating the joint PDF over the other variable
  • Marginal PDF of $X$: $f_X(x) = \int_{-\infty}^{\infty}f_{X,Y}(x,y)dy$
  • Marginal PDF of $Y$: $f_Y(y) = \int_{-\infty}^{\infty}f_{X,Y}(x,y)dx$
  • Allows for analyzing individual random variables separately

Interpretation of joint density graphs

• Joint PDFs visually represent the probability distribution of two or more continuous random variables together
• 3D surface plot interpretation
  • Height of the surface at any point $(x,y)$ represents the joint PDF value $f_{X,Y}(x,y)$ at that point
  • Higher surface points indicate more likely combinations of $x$ and $y$ values
• Contour plot interpretation
  • Lines of constant probability density in the $xy$-plane
  • Closer contour lines indicate steeper changes in probability density
• Regions with higher joint PDF values signify where the random variables are more likely to take on those specific values simultaneously (peaks in the surface plot or dense contour lines)

Calculations with joint density functions

• Probability calculation: the probability of two continuous random variables falling within a specific region $R$ is the double integral of the joint PDF over that region
  • $P((X,Y) \in R) = \int\int_R f_{X,Y}(x,y)dxdy$
  • Rectangular region example: $P(a \leq X \leq b, c \leq Y \leq d) = \int_c^d\int_a^b f_{X,Y}(x,y)dxdy$
• Conditional PDFs: calculated using the joint PDF and the marginal PDF
  • $f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_X(x)}$, where $f_X(x) \neq 0$
  • Allows for analyzing the distribution of one variable given a specific value of the other

Double integrals for joint densities

• Double integrals calculate probabilities, expected values, and other quantities related to joint PDFs
• Expected value calculation: the expected value of a function $g(X,Y)$ is
  • $E[g(X,Y)] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(x,y)f_{X,Y}(x,y)dxdy$
  • Weighted average of $g(X,Y)$ over all possible values of $X$ and $Y$
• Covariance calculation: measures the linear relationship between two random variables $X$ and $Y$
  • $Cov(X,Y) = E[(X-E[X])(Y-E[Y])] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x-\mu_X)(y-\mu_Y)f_{X,Y}(x,y)dxdy$
  • $\mu_X = E[X]$ and $\mu_Y = E[Y]$ are the means of $X$ and $Y$
• Correlation coefficient calculation: standardized measure of the linear relationship between $X$ and $Y$
  • $\rho_{X,Y} = \frac{Cov(X,Y)}{\sigma_X\sigma_Y}$, where $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$
  • Values range from -1 (perfect negative correlation) to 1 (perfect positive correlation), with 0 indicating no linear correlation