Question

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 |
||||

,
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 a cycle?

(b) What is the probability that there is at most one car and at
most one bus during a cycle?

(c) What is the probability that there is exactly one car during a
cycle? Exactly one bus?

P(exactly one car) |
= |

P(exactly one bus) |
= |

(d) Suppose the left-turn lane is to have a capacity of five cars
and one bus is equivalent to three cars. What is the probability of
an overflow during a cycle?

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

Yes, because *p*(*x, y*) =
*p*_{X}(*x*) ·
*p*_{Y}(*y*).Yes, because
*p*(*x, y*) ≠
*p*_{X}(*x*) ·
*p*_{Y}(*y*). No,
because *p*(*x, y*) =
*p*_{X}(*x*) ·
*p*_{Y}(*y*).No, because
*p*(*x, y*) ≠
*p*_{X}(*x*) ·
*p*_{Y}(*y*).

Answer #1

5.1) 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...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.125
0.075
0.050
3
0.150
0.090
0.060
4
0.100
0.060...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.085
0.075
0.050
3
0.150
0.090
0.060
4
0.100
0.060...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.050
0.075
0.050
3
0.150
0.090
0.060
4
0.100
0.060...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.110
0.075
0.050
3
0.150
0.090
0.060
4
0.100
0.060...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.105
0.075
0.050
3
0.150
0.090...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.105
0.075
0.050
3
0.150
0.090...

Find the joint discrete random variable x and y,their joint
probability mass function is given by
Px,y(x,y)={k(x+y);x=-2,0,+2,y=-1,0,+1 K>0 0
Otherwise } 2.1 determine the value of constant k,such
that this will be proper pmf? 2.2 find the marginal pmf’s,Px(x) and
Py(y)? 2.3 obtain the expected values of random variables X and Y?
2.4 calculate the variances of X and Y?
Hint:££Px,y(x,y)=1,Px(x)=£Px,y(x,y);Py(y)=£Px,y(.
x,y);E[]=£xpx(x);

Find the joint discrete random variable x and y,their joint
probability mass function is given by Px,y(x,y)={k(x+y)
x=-2,0,+2,y=-1,0,+1
0 Otherwise }
2.1 determine the value of constant k,such that this will be
proper pmf?
2.2 find the marginal pmf’s,Px(x) and Py(y)?
2.3 obtain the expected values of random variables X and Y?
2.4 calculate the variances of X and Y?

Random variables X and Y assume values 1, 2 and 3.
Their joint probability distribution is given as follows:
P(X=Y=1) = min( 0.39, 0.19 ) ,
P(X=Y=2) = min( 0.19, 0.42 ) and
P(X=Y=3) = min( 0.42 0.39, ), ....
Their marginal probability distributions are as follows:
P(X=1) = 0.39, P(Y=1) = 0.19,
P(X=2) = 0.19, P(Y=2) = 0.42,
P(X=3) = 0.42 and PY=3) = 0.39,
Calculate the variance of the sum (X + Y)

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