Joint Distributions and Functions of Random Variables
This domain covers joint distributions, marginal and conditional distributions, covariance and correlation, and deriving the distribution of a function of random variables. CS1 expects both the convolution approach and the Jacobian transformation method.
Core rules
- Joint density f_{X,Y}(x,y) integrates to 1; marginals integrate out the other variable.
- Independence iff f_{X,Y}(x,y)=f_X(x)f_Y(y) (or the cdf factorises).
- Cov(X,Y)=E[XY]−E[X]E[Y]; Corr is Cov divided by the standard deviations. Independence implies zero correlation, not conversely.
- For U=g(X,Y), V=h(X,Y), use f_{U,V}(u,v)=f_{X,Y}(x(u,v),y(u,v))|J|, where J is the Jacobian determinant of the inverse transformation.
- For sums, convolve: f_{X+Y}(z)=∫f_X(x)f_Y(z−x)dx over the valid support.
Worked example
Let X,Y be independent Uniform(0,1). For Z=X+Y, f_Z(z)=∫0^1 1·1{0<z−x<1} dx. For 0<z<1, the integrand is 1 for 0<x<z, giving f_Z(z)=z. For 1≤z<2, it is 1 for z−1<x<1, giving f_Z(z)=2−z. Outside (0,2), f_Z=0. Checking ∫_0^1 z dz+∫_1^2(2−z)dz=1 confirms the triangular density.
Traps
- Losing the support: the convolution limits change at z=1.
- Omitting the absolute value of the Jacobian, or using the forward rather than inverse map.
- Concluding independence from zero correlation.
- For discrete variables, summing over the correct finite range rather than integrating.