Math 408, Actuarial Statistics I A.J. HildebrandJoint Distributions, Continuous CaseIn the following, X and Y are continuous random variables. Most of the concepts and formulasbelow are analogous to those for the discrete case, with integrals replacing sums. The principaldifference between continuous lies in the definition of the p.d.f./p.m.f. f (x, y): The formula f (x, y) =P (X = x, Y = y) is no longer valid, and there is no simple and direct way to obtain f (x, y) from Xand Y .1. Joint continuous distributions:• Joint density (joint p.d.f.): A function f(x, y) satisfying (i) f(x, y) ≥ 0, (ii)RRf(x, y)dxdy =1. Usually, f(x, y) will be given by an explicit formula, along with a range (a region inthe xy-plane) on which this formula holds. In the general formulas below, if a range ofintegration is not explicitly given, the integrals are to be taken over the range in whichthe density function is defined.• Uniform joint distribution: An important special type of joint density is one that isconstant over a given range (a region in the xy-plane), and 0 outside outside this range,the constant being the reciprocal of the area of of the range. This is analogous to theconcept of an ordinary (one-variable) uniform density f(x) over an interval I, which isconstant (and equal to the reciprocal of the length of I) inside the interval, and 0 outsideit.2. Marginal distributions: The ordinary distributions of X and Y , when considered sepa-rately. The corresponding (one-variable) densities are denoted by fX(or f1) and fY(or f2),and obtained by integrating the joint density f (x, y) over the “other” variable:fX(x) =Zf(x, y)dy, fY(y) =Zf(x, y)dx.3. Computations with joint distributions:• Probabilities:Given a region R in the xy-plane the probability that (X, Y ) falls into this region is givenby the double integral of f (x, y) over this region. For example, P (X + Y ≤ 1) is givenby an integral of the formRRRf(x, y)dxdy, where R consists of the part of the range off in which x + y ≤ 1.• Expectation of a function of X and Y (e.g., u(x, y) = xy):E(u(X, Y )) =RRu(x, y)f (x, y)dxdy4. Covariance and correlation:The formulas and definitions are the same as in the discrete case.• Definitions: Cov(X, Y ) = E(XY ) − E(X)E(Y ) = E((X − µX)(Y − µY)) (Covarianceof X and Y ), ρ = ρ(X, Y ) =Cov(X,Y )σXσY(Correlation of X and Y )• Properties: | Cov(X, Y )| ≤ σXσY, −1 ≤ ρ(X, Y ) ≤ 1• Relation to variance: Var(X) = Cov(X, X)• Variance of a sum: Var(X + Y ) = Var(X) + Var(Y ) + 2 Cov(X, Y )5. Independence of random variables: Same as in the discrete case:• Definition: X and Y are called independent if the joint p.d.f. is the product of theindividual p.d.f.’s: i.e., if f(x, y) = fX(x)fY(y) for all x, y.1Math 408, Actuarial Statistics I A.J. Hildebrand• Properties of independent random variables:If X and Y are independent, then:– The expectation of the product of X and Y is the product of the individualexpectations: E(XY ) = E(X)E(Y ). More generally, this product formula holdsfor any expectation of a function X times a function of Y . For example, E(X2Y3) =E(X2)E(Y3).– The product formula holds for probabilities of the form P(some condi-tion on X, some condition on Y ) (where the comma denotes “and”): Forexample, P (X ≤ 2, Y ≤ 3) = P (X ≤ 2)P (Y ≤ 3).– The covariance and correlation of X and Y are 0: Cov(X, Y ) = 0, ρ(X, Y ) = 0.– The variance of the sum of X and Y is the sum of the individual variances:Var(X + Y ) = Var(X) + Var(Y )– The moment-generating function of the sum of X and Y is the product ofthe individual moment-generating functions: MX+Y(t) = MX(t)MY(t).6. Conditional distributions: Same as in the discrete case, with integrals in place of sums:• Definitions:– conditional density of X given that Y = y:g(x|y) =f(x,y)fY(y )– conditional density of Y given that X = x:h(y|x) =f(x,y)fX(x)• Conditional expectations and variance: Conditional expectations, variances, etc.,are defined and computed as usual, but with conditional distributions in place of ordinarydistributions. For example:• E(X|Y = 1) = E(X|Y = 1) =Rxg(x|1)dx• E(X2|Y = 1) = E(X2|Y = 1)
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