Question

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation. y p(x, y) 0 1 2 x 0 0.10 0.05 0.01 1 0.07 0.20 0.07 2 0.05 0.14 0.31

(a) What is P(X = 1 and Y = 1)?

P(X = 1 and Y = 1) =

(b) Compute P(X ≤ 1 and Y ≤ 1).

P(X ≤ 1 and Y ≤ 1) =

(c) Give a word description of the event {X ≠ 0 and Y ≠ 0}. At most one hose is in use at both islands. One hose is in use on both islands. One hose is in use on one island. At least one hose is in use at both islands. Compute the probability of this event. P(X ≠ 0 and Y ≠ 0) =

(d) Compute the marginal pmf of X. x 0 1 2 pX(x) Compute the marginal pmf of Y. y 0 1 2 pY(y) Using pX(x), what is P(X ≤ 1)? P(X ≤ 1) =

(e) Are X and Y independent rv's? Explain.

X and Y are independent because P(x,y) = pX(x) · pY(y).

X and Y are not independent because P(x,y) ≠ pX(x) · pY(y).

X and Y are independent because P(x,y) ≠ pX(x) ·

pY(y). X and Y are not independent because P(x,y) = pX(x) · pY(y).

Answer #1

A service station has both self-service and full-service
islands. On each island, there is a single regular unleaded pump
with two hoses. Let X denote the number of hoses being
used on the self-service island at a particular time, and let
Y denote the number of hoses on the full-service island in
use at that time. The joint pmf of X and Y
appears in the accompanying tabulation.
y
p(x,
y)
0
1
2
x
0
0.10
0.03
0.02 ...

A service station has both self-service and full-service
islands. On each island, there is a single regular unleaded pump
with two hoses. Let X denote the number of hoses being used on the
self-service island at a particular time, and let Y denote the
number of hoses on the full-service island in use at that time. The
joint pmf of X and Y appears in the accompanying tabulation. y p(x,
y) 0 1 2 x 0 0.10 0.03 0.01 1...

A service station has both self-service and full-service
islands. On each island, there is a single regular unleaded pump
with two hoses. Let X denote the number of hoses being
used on the self-service island at a particular time, and let
Y denote the number of hoses on the full-service island in
use at that time. The joint pmf of X and Y
appears in the accompanying tabulation.
y
p(x,
y)
0
1
2
x
0
0.10
0.03
0.01 ...

A service station has both self-service and full-service
islands. On each island, there is a single regular unleaded pump
with two hoses. Let X denote the number of hoses being used on the
self-service island at a particular time, and let Y denote the
number of hoses on the full-service island in use at that time. The
joint pmf of X and Y appears in the accompanying tabulation.
P(X=x,Y=y) Y X 0 1 2 0 0.1 0.04 0.02 1 0.08...

The joint probability distribution of the number X of
cars and the number Y of buses per signal cycle at a
proposed left-turn lane is displayed in the accompanying joint
probability table.
y
p(x,
y)
0
1
2
x
0
0.010
0.015
0.025
1
0.020
0.030
0.050
2
0.050
0.075
0.125
3
0.060
0.090
0.150
4
0.040
0.060
0.100
5
0.020
0.030
0.050
(a) What is the probability that there is exactly one car and
exactly one bus during...

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1. Indicate if each of the following is true or false. If false,
provide a counterexample.
(a) The mean of a sample is always the same as the median of
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(b) The mean of a population is the same as that of a
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For problem 1, to compute probabilities, the probability of
event 1 AND event 2 is obtained by multiplying the two
probabilities together, whereas the probability of event 1 OR event
2 is obtained by adding the two probabilities.
1. Duchenne muscular
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