Joint probability density
Definition taught in Joint distributions, part of Probability
Joint probability density
| Type | Definition |
| Taught in | Joint distributions |
| Part | Probability |
| Properties | 1 |
| Worked examples | 2 |
| Questions using it | 17 |
Continuous random variables X and Y have joint density f when, for every reasonable region B,
\mathbb{P}((X,Y)\in B)=\iint_B f(x,y)\,\,\mathrm{d} x\,\,\mathrm{d} y.
The marginal density of X is f_X(x)=\int f(x,y)\,\,\mathrm{d} y.
Properties
Transformation rule
Main article: Transformation rule
Let \mathbf{X} take values in an open set of \mathbb{R}^n with density f_{\mathbf{X}}, and let \mathbf{Z}=g(\mathbf{X}) where g is a smooth bijection onto its image, with inverse g^{-1} and Jacobian determinant J that does not vanish. The density of \mathbf{Z} is
f_{\mathbf{Z}}(\mathbf{z})=f_{\mathbf{X}}\bigl(g^{-1}(\mathbf{z})\bigr)\,\bigl|\det Dg^{-1}(\mathbf{z})\bigr|.
Equivalently, if the Jacobian is computed for g rather than for the inverse, divide by |\det Dg| evaluated at the preimage. For a linear map \mathbf{Z}=A\mathbf{X} with A invertible, f_{\mathbf{Z}}(\mathbf{z})=f_{\mathbf{X}}(A^{-1}\mathbf{z})/|\det A|.
Examples
A probability as an integral
Question page: A probability as an integral
Explain how to calculate
\mathbb{P}\Bigl(\frac{X_1^2+\sin X_2}{X_1^2 X_2}>1\Bigr)
from the joint density of (X_1,X_2). Assume the expression under the probability is defined.
Integrate the joint density over the region where (x_1^2+\sin x_2)/(x_1^2 x_2)>1.
A product density
Question page: A product density
A pair (X,Y) has joint density f(x,y)=2e^{-x-2y} for x>0 and y>0.
Calculate \mathbb{E}(XY).
Calculate \operatorname{Cov}(X+Y, X-Y).
The joint density factors as e^{-x}\cdot 2e^{-2y} on the positive quadrant, so X\sim\operatorname{Exp}(1) and Y\sim\operatorname{Exp}(2) are independent.
\mathbb{E}(XY)=(\mathbb{E} X)(\mathbb{E} Y)=1\times 1/2=1/2.
The cross covariances cancel:
\operatorname{Cov}(X+Y,X-Y)=\operatorname{Var}(X)-\operatorname{Var}(Y)=1-\frac14=\frac34.
Used in
- Taught in Joint distributions, where the derivation and examples are.
- Also used in Functions of random variables and limit theorems · Statistical inference · Estimation · Bayesian foundations · Conjugate analysis for normal models · Markov chains, Gibbs sampling, and Metropolis–Hastings · Bayesian hypotheses and model comparison · Exponential families and conjugacy · Hierarchical Bayes
- Questions that use it (17) — A probability as an integral · A product density · A point in a triangle · Sum and proportion of two exponentials · A bivariate-normal Gibbs sampler (answer) · A normal statistic · Q4 · Q9 (answer) · Q18 (answer) · Q23 (answer) · Q25 · Q26 · Q58 (answer) · Question 6 · Question 9 · Question 16 (answer) · Question 17
- All 8 questions on Joint distributions
See also
Random variable · Bayes’ rule · Exchangeability · Independent random variables · Random sample · Systematic-scan Gibbs sampler · Joint probability mass function · Covariance