Description
-
To win a prize at the fair, you need to hit the bullseye once with a bow and arrow. Your probability of
success is only p = 0.2 per shot. You get three shots; if you hit the bullseye, you stop shooting. Let X
be the number of shots taken, and Y indicate winning the prize (Y = 1 for success, Y = 0 for failure.)
X and Y have joint PMF:
PX,Y x=1 x=2 x=3
y = 0 0.00 0.00 0.512
y = 1 0.20 0.16 0.128
- (a) Calculate PX,Y|B(x,y), where B = {X > 1}. What “experiment” does this describe?
- (b) Calculate E[X|Y = 1].
- (c) CalculatePY|X(y|2)andexplainit’sbehavior.
- (d) Calculate σX|Y=1 and σX|Y=0 and interpret their values.
- (e) Suppose you didn’t know the probability p. If you were given E[X], could you determine p from it? What about if you were given E[X|Y = 0] or E[X|Y = 1]?
- X is a discrete random variable. Let Y = aX +b. Show that if a > 0, ρ(X,Y) = 1, and that if a < 0, ρ(X,Y) = −1.
-
Prove that |ρ(X,Y)| ≤ 1 for any (discrete) X and Y.
Hint: calculate the variance of Z = X ± Y , and use the fact that variance must be nonnegative.
σX σY
- Domino’s Pizza delivery times (X) are continuously and uniformly distributed between 30 and 40 minutes. Little Caesar’s Pizza delivery times (Y ) are continuously and uniformly distributed between 20 and 50 minutes. X and Y are independent, so that fX ,Y (x, y) = fX (x) fY (y). Suppose you simultaneously order from both. What is P(X < Y ), the probability that Domino’s delivers first?
-
Suppose, at your bus stop, “local” buses have a mean wait time of 10 min, such that the time you wait
for a local bus L is an exponential random variable with μL = 10 min. “Express” buses have a mean
wait time of 30 min; the express-bus wait time X is an exponential random variable with
μX = 30 min. The bus schedules are not coordinated, so X and L are independent.- (a) What is the joint PDF fL,X(l,x)?
- (b) What is the probablility that the first bus to arrive is a local one; that is, P(L < X )?
- (c) Suppose the travel time to your job is 30 minutes on the local bus, and 20 minutes on the express bus. If you always get on the first bus to arrive, what is the expected time of your commute, include time spent waiting at the bus station?
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