Unformatted text preview: 27 Conditional Distributions  Recall Section 1.7. Let A and B be any events in 8. Suppose A depends on B7 knowing
B adds information about A. The conditional probability of A given B is denoted
P(AjBi Pom B) P(A P(B‘) 3) a Let X and Y be continuous random variables with joint pdf fXJ(Jr, y). The conditional
distribution of X given Y : y is defined by fX,1"(:Li y)
f3" (y) for which the rules about mean and variance apply. it (IY :y) : o The conditional expected value of X given Y : y is “XI?” : E[XfY=yl
: ] :zfxifi’:y)dr
XEX o The conditional variance of X given 1’ :_ y is “in? : vai'lXiYZEl =/ (inﬂxiyfﬁii
.rEX Y:y)d;t and can be found by vmgxly : y] : E [my : MEE my : y]  Example: A soft drink machine has a random amount X in supply (meeeured in
hundreds of gallons) at the beginning of a given day and dispenses a random amount
Y during the day (measured in hundreds of gallons). Clearly, Y S X. X and Y have
a joint distribution given by x/‘v ﬂX,Y):%I<0snys2) (a) \Viiat IS the marginal distiibution of the amount of soft drink in supply? X 3%? stem/iii in Sopf’i/
)4 {he mom if'iefﬂnﬂ
)ﬁfx .41 59 oé ...
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 Fall '07
 Carlton

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