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assignment_3_08

# assignment_3_08 - ESO 209 PROBABILITY STATISTICS Semester 2...

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ESO 209: PROBABILITY & STATISTICS Semester 2: 2007-08 Assignment #3 Instructor: Amit Mitra [1] Let be a random variable defined on X ( ) , P Ω F, . Show that the following are also random variables; (a) || , (b) and (c) X 2 X X , given that { } 0 X φ <= . [2] Let [ ] 0,1 Ω= and be the Borel F σ field of subsets of Ω . Define on X Ω as follows: () if 0 1 2 12 if 12 1 X ωω ω ≤≤ = <≤ Show that defined above is a random variable. X [3] Let a card be selected from an ordinary pack of playing cards. The outcome is one of these 52 cards. Define on X Ω as: 4i f i s a n a c e 3 if is a king 2 if is a queen 1 if is a jack 0o t h e r w i s e . X = Show that is a random variable. Further, suppose that X ( ) . P assigns a probability of 152 to each outcome . Describe the induced probability ( ) .. X P [4] Let () ( ) 0i f 1 24i f - 1 1 1i f 1 . x Fx x x x < − = +≤ < Show that is a distribution function. Sketch the graph of and compute the probabilities (a) . F ( ) PX −< , (b) ( ) 0 = , (c) ( ) 1 = and (d) ( ) 11 −≤ < .

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assignment_3_08 - ESO 209 PROBABILITY STATISTICS Semester 2...

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