Stat 5101 Lecture Notes - School of Statistics

3.5. CONDITIONAL EXPECTATION AND PREDICTION 105Of course, the joint is conditional times marginalf(x, y) =f(y|x)f X (x)=xe −xy · λe −λx = λxe −(λ+y)x (3.32)Question: What is the other marginal (of Y ) and the other conditional (ofX given Y )? Note that these two problems are related. If we answer one, theanswer to the other is easy, just a divisionf(x | y) =f(x, y)f Y (y)orf(x, y)f Y (y) =f(x | y)I find it a bit easier to get the conditional first. Note that the joint (3.32) is anunnormalized conditional when thought of as a function of x alone. Checkingour inventory of “brand name” distributions, we see that the only one like(3.32) in having both a power and an exponential of the variable is the gammadistribution with densityf(x | α, λ) =λαΓ(α) xα−1 e −λx , x > 0. (3.33)Comparing the analogous parts of (3.32) and (3.33), we see that we must matchup x with x α−1 , which tells us we need α = 2, and we must match up e −(λ+y)xwith e −λx which tells us we need λ + y in (3.32) to be the λ in (3.33), whichis the second parameter of the gamma distribution. Thus (3.32) must be anunnormalized Γ(2,λ+y) density, and the properly normalized density isf(x | y) =(λ+y) 2 xe −(λ+y)x , x > 0 (3.34)Again this is a bit tricky, so let’s go through it slowly. We want to change α to2 and λ to λ + y in (3.33). That givesf(x | y) = (λ+y)2 x 2−1 e −(λ+y)x , x > 0.Γ(2)and this cleans up to give (3.34).3.5 Conditional Expectation and PredictionThe parallel axis theorem (Theorem 2.11 in these notes)E[(X − a) 2 ] = var(X)+[a−E(X)] 2has an analog for conditional expectation. Just replace expectations by conditionalexpectations (and variances by conditional variances) and, because functionsof the conditioning variable behave like constants, replace the constant bya function of the conditioning variable.