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Unformatted text preview: ribution fY |X (v |uo ) is not deﬁned
for other values of uo . The conditional pdfs are shown in Figure 4.8. 4.3. JOINT PROBABILITY DENSITY FUNCTIONS 129 Example 4.3.4 Suppose X and Y have the joint pdf:
e−u if 0 ≤ v ≤ u
0 else. fX,Y (u, v ) = (4.10) The pdf and its support are shown in Figure 4.9. Find the marginal and conditional pdfs.
f v (u,v) v X,Y support
u u Figure 4.9: The pdf (4.10) and its support. Solution: uo −uo
0e fX (uo ) = = uo e−uo uo ≥ 0
else. 0 fY (vo ) = ∞ −u
vo e = e−vo 0 vo ≥ 0
else. The pdfs are shown in Figure 4.10.
f (u) f (v) X Y u f Y|X v (v|uo ) f X|Y (u|vo ) v u
vo uo Figure 4.10: The marginal and conditional pdfs for the joint pdf in (4.10).
The conditional density fY |X (v |uo ) is undeﬁned if uo ≤ 0. For uo > 0 :
fY |X (v |uo ) = e−uo
uo e−uo 0 = 1
uo 0 ≤ v ≤ uo
else. 130 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES That is, the conditional distribution of Y given X = uo is the uniform distribution over the interval
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
- Spring '08
- The Land