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EECS 501 EXAM #1 Fall 2001
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(40) 1. Random variables x, y have joint pdf fx,y (X, Y ) =
cXY if 0 < Y < X < 1 0 otherwise where c is a constant. Random variable z = y/x. (05) a. Compute the constant c in the pdf fx,y (X, Y ). (05) b. Are x and y independent? Explain your answer. (05) c. Compute the marginal pdf fx(X). (05) d. Compute the conditional pdf fy|x(Y |X) at X = 1/2. (10) e. Compute the pdf fz (Z) using the method of events. (10) f. Compute P r[(x + y) < 1]. Hint: inner integral over y.
NOTE: Half-credit if you do this problem with “cXY” replaced with “c” in fx,y (X, Y ).
WRITE YOUR ANSWERS HERE. SIMPLIFY TO A FRACTION.
(a): (c): (e):
(b): (d): (f ):
(40) 2. We flip coin A, which has Pr[heads]=2/3. All flips are independent. If coin A lands heads, we flip coin B, which has Pr[heads]=3/4. If coin A lands tails, we flip coin C, which has Pr[heads]=4/5. (05) a. Compute Pr[the second coin flipped (whatever it is) lands heads]. (05) b. Compute Pr[Coin A landed heads|second coin flipped lands heads].
Now the second coin (whatever it is) is flipped n − 1 more times (total of n flips).
(05) c. Compute Pr[All n flips of the second coin (whatever it is) land heads]. (05) d. Compute Pr[Coin A landed heads|all n flips of second coin land heads]. (05) e. Compute (^) nLIM→∞ [your answer to (d)]. You don’t need to be rigorous. (05) f. Interpret your answer to (e): Explain why it (hopefully!) makes sense.
(10) g. PROVE that if events E and F are independent, then events E and F ′^ are also independent, where F ′=set complement of F. HINT: Problem Set #1.
WRITE YOUR ANSWERS HERE. SIMPLIFY TO A FRACTION.
(a): (c):
(b): (d):
(e): (f ):
(g):
