Question
In: Statistics and Probability
1. A service station has both self-service and full-service islands. On each island, there is a...
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 par- ticular time, and let Y denote the number of hoses on the full service island in use that time. The joint pmf of X and Y appears in the accompanying tabulation.
| 0 | 1 | 2 | |
| 0 | .10 | .04 | .02 |
| 1 | .08 | .20 | .06 |
| 2 | .06 | .14 | .30 |
a. WhatisP(X=1andY =1)?
b. ComputeP(X≤1andY ≤1).
c. Compute P(X ̸= 1 and Y ̸= 1).
d. Compute the marginal pmf of X and Y. Using pX(x), what is P(X ≤ 1)?
e. Are X and Y are independent? Explain.
2. Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable – X for the right tire and Y for the left tire, with joint pdf
?K(x2 +y2), 20≤x≤30, 20≤y≤30,
f(x,y) =
0, otherwise
a. What is the value of K?
b. What is the probability that both tires are underfilled?
c. What is the probability that the difference in air pressure between the two tires is at most 2 psi? (Hint: Draw the shaded region first and then find the boundary of the integration.)
d. Determine the marginal pdf of X and Y . e. Are X and Y
independent? Explain.
f. Find E(X) and E(Y ).
Solutions
orchestra answered 1 year agoRelated Solutions
A service station has both self-service and full-service islands. On each island, there is a single...
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 0.02 1...
A service station has both self-service and full-service islands. On each island, there is a single...
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...
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
0.02 ...
A service station has both self-service and full-service islands. On each island, there is a single...
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 ...
A service station has both self-service and full-service islands. On each island, there is a single...
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,y)
y
x
0
1
2
0
0.06
0.14
0.02
1
0.03...
A service station has both self-service and full-service islands. On each island, there is a single...
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 ...
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