联合概率分布：离散与连续随机变量.pdf

Math 370/408, Actuarial Problemsolving A.J. Hildebrand Joint Distributions, Discrete Case In the following, X and Y are discrete random variables. 1. Joint distribution (joint p.m.f.): • Definition: f (x, y) = P (X = x, Y = y) • Properties: (1) f (x, y) ≥ 0, (2) f (x, y) = 1 x,y • Representation: The most natural representation of a joint discrete distribution is as a distribution matrix, with rows and columns indexed by x and y , and the xy-entry being f (x, y). This is analogous to the representation of ordinary discrete distributions as a single-row table. As in the one-dimensional case, the entries in a distribution matrix must be nonnegative and add up to 1. 2. Marginal distributions: The distributions of X and Y , when considered separately. • Definition: • fX (x) = P (X = x) = f (x, y) y • fY (y) = P (Y = y) = f (x, y) x • Connection with distribution matrix: The marginal distributions fX (x) and fY (y) can be obtained from the distribution matrix as the row sums and column sums of the entries. These sums can be entered in the “margins” of the matrix as an additional column and row. • Expectation and variance: µX , µY , σ2 , σ2 denote the (ordinary) expectations and X Y variances of X and Y , computed as usual: µX = xfX (x), etc. x 3. Computations with joint distributions: • Probabilities: Pr