Section 8 – Joint, Marginal, and Conditional Distributions

Joint Distribution of X and Y

CDF of a Joint Distribution

Expectation of a function of Jointly Distributed RV’s Recall: Now:

Marginal Distribution Formulas Caution: When the probability space is non- rectangular, make sure to set limits of integration correctly – Example 8-9 in Actex

Independent RV’s: X & Y

Marginal Distribution of X and Y Before this chapter, we were dealing with one random variable – These RV’s had f(x)  this was a marginal distribution for X – fx(X) = Probability that that value of X occurs This is what we’ve already been doing! Coin-dice example: – X: coin toss (Tails=0, Heads=1) – If get heads roll 1 die, if tails roll 2 – Y=Total number rolled on dice – fx(0) =.5, fx(1) =.5 – fy(1) = fx,y(1,1)=(.5)(1/6)  there is no possible y= 1 if x = 0 – fy(2) = fx,y(0,2) + fx,y(1,2) = (.5)(1/36)+(.5)(1/6) Sum (over all x) any events where y = 2

Conditional Distribution of Y given X=x

Expectation & Variance of Conditional Expectation – Find the conditional PDF from previous formula – Apply the expectation formula like usual Variance (trickier!) – Find the conditional PDF from previous formula – Apply the expectation formula like usual Find conditional mean: E[Y|X=x) Find conditional second moment: E[Y^2|X=x] Use the variance formula like usual, using these components

Two Formulas for f(x,y) For any X, Y Special case for when X, Y are independent

Covariance between X & Y Covariance = 0 for independent X, Y Positive for large X with large Y Negative for large X with small Y (vice versa) Formula is similar to our familiar variance formula

Moment Generating Function of a Joint Distribution The E(e^tX) and M’(0) approaches both work Can get E(XY), E(X), E(Y) from this  Cov(X,Y)

Bivariate Normal Distribution “The Bivariate Normal Distribution has occurred infrequently on Exam P More information in Actex (p. 236)

Properties If X and Y are independent: Products of expectations are expectations of products – E[g(X) * h(Y)]=E[g(X)] * E[h(Y)] Particularly useful: E[XY] = E[X] * E[Y] This is why Cov(X,Y) = 0 when X&Y are independent Cov[aX + bY + c, dZ + eW + f] = adCov[X,Z] + aeCov[X,W] + bdCov[Y,Z] + beCov[Y,W] Lower case constants, upper case RV’s Similer to the foil method…i.e.(a+b)(c+d)=ac+ad+bc+bd Var[aX + bY + c] =a 2 *Var[X] + b 2 *Var[Y] + 2ab*Cov[X,Y] Remember:The sign of a, b affect Cov term

Formulas to Understand Graphically Make sure that you understand these formulas as graphical concepts – Be able to set up these problems by thinking instead of memorizing

There’s some real work to do! From here on out, STAT 414 is really not enough Go through all of the examples in the chapter and be comfortable setting up double integrals to find probabilities Understand all of the properties/formulas – There are more properties on p than what I covered This is some of the most conceptually difficult material on the entire exam that is frequently tested Practice problems! Bring questions for next time!