Question

An instructor has given a short quiz consisting of two parts.
For a randomly selected student, let *X* = the number of
points earned on the first part and *Y* = the number of
points earned on the second part. Suppose that the joint pmf of
*X* and *Y* is given in the accompanying table.

y |
|||||

,
x)y |
0 | 5 | 10 | 15 | |

x |
0 | 0.03 | 0.06 | 0.02 | 0.10 |

5 | 0.04 | 0.17 | 0.20 | 0.10 | |

10 | 0.01 | 0.15 | 0.11 | 0.01 |

(a) Compute the covariance for *X* and *Y*. (Round
your answer to two decimal places.)

Cov(*X*, *Y*) =

(b) Compute *ρ* for *X* and *Y*. (Round your
answer to two decimal places.)

*ρ* =

Answer #1

from given data:

marginal distribution of x:

x | P(x) | xP(x) | x^2P(x) |

0 | 0.210 | 0.000 | 0.000 |

5 | 0.510 | 2.550 | 12.750 |

10 | 0.280 | 2.800 | 28.000 |

total | 1.000 | 5.350 | 40.750 |

E(x) | = | 5.3500 | |

E(x^2) | = | 40.7500 | |

Var(x)= | E(x^2)-(E(x))^2= | 12.1275 |

marginal distribution of Y:

y | P(y) | yP(y) | y^2P(y) |

0 | 0.080 | 0.000 | 0.000 |

5 | 0.380 | 1.900 | 9.500 |

10 | 0.330 | 3.300 | 33.000 |

15 | 0.210 | 3.150 | 47.250 |

total | 1.000 | 8.350 | 89.750 |

E(y) | = | 8.3500 | |

E(y^2) | = | 89.7500 | |

Var(y)=σy= | E(y^2)-(E(y))^2= | 20.0275 |

E(XY) =41.75

**a)**

Covar(y,x)=E(XY)-E(X)*E(Y)= |
-2.9225 |

**b)**

Correlation coefficient ρ=Cov(X,Y)/√(σx*σy)= |
-0.1875 |

An instructor has given a short quiz consisting of two parts.
For a randomly selected student, let X = the number of points
earned on the first part and Y = the number of points earned on the
second part. Suppose that the joint pmf of X and Y is given in the
accompanying table. y p(x, y) 0 5 10 15 x 0 0.03 0.06 0.02 0.10 5
0.04 0.14 0.20 0.10 10 0.01 0.15 0.14 0.01 (a) Compute...

An instructor has given a short quiz consisting of two parts.
For a randomly selected student, let X = the number of
points earned on the first part and Y = the number of
points earned on the second part. Suppose that the joint pmf of
X and Y is given in the accompanying table.
y
p(x, y)
0
5
10
15
x
0
0.03
0.06
0.02
0.10
5
0.04
0.15
0.20
0.10
10
0.01
0.15
0.13
0.01
(a)...

An instructor has given a short quiz consisting of two parts.
For a randomly selected student, let X = the number of
points earned on the first part and Y = the number of
points earned on the second part. Suppose that the joint pmf of
X and Y is given in the accompanying table.
y
p(x,
y)
0
5
10
15
x
0
0.03
0.06
0.02
0.10
5
0.04
0.16
0.20
0.10
10
0.01
0.15
0.12
0.01
(a)...

An instructor has given a short quiz consisting of two parts.
For a randomly selected student, let X = the number of
points earned on the first part and Y = the number of
points earned on the second part. Suppose that the joint pmf of
X and Y is given in the accompanying table.
y
p(x,
y)
0
5
10
15
x
0
0.03
0.06
0.02
0.10
5
0.04
0.15
0.20
0.10
10
0.01
0.15
0.13
0.01
(a)...

An instructor has given a short quiz consisting of two parts.
For a randomly selected student, let X = the number of
points earned on the first part and Y = the number of
points earned on the second part. Suppose that the joint pmf of
X and Y is given in the accompanying table.
y
p(x,
y)
0
5
10
15
x
0
0.01
0.06
0.02
0.10
5
0.04
0.14
0.20
0.10
10
0.01
0.15
0.16
0.01
(a)...

An instructor has given a short quiz consisting of two parts.
For a randomly selected student, let X represent the number of
points earned on the first part and let Y represent the number of
points earned on the second part. Suppose that the joint pmf of X
and Y is given in the table below. Y 0 5 10 15 0 0.11 0.07 0.05
0.12 X 5 0.07 0.14 0.09 0.07 10 0.06 0.10 0.04 0.08 (a) Calculate
P(X...

Consider joint Probability distribution of two random variables
X and Y given as following
f(x,y) X
2 4 6
Y 1 0.1 0.15
0.06
3 0.17 0.1
0.18
5 0.04 0.07
0.13
(a) Find expected value of g(X,Y) = XY2
(b) Find Covariance of Cov(x,y)

A certain market has both an express checkout line and a
super-express checkout line. Let X1 denote the
number of customers in line at the express checkout at a particular
time of day, and let X2 denote the number of
customers in line at the superexpress checkout at the same time.
Suppose the joint pmf of X1 and
X2 is as given in the accompanying table.
x2
0
1
2
3
x1
0
0.08
0.06
0.04
0.00
1
0.04
0.18...

A certain market has both an express checkout line and a
super-express checkout line. Let X1 denote the
number of customers in line at the express checkout at a particular
time of day, and let X2 denote the number of
customers in line at the superexpress checkout at the same time.
Suppose the joint pmf of X1 and
X2 is as given in the accompanying table.
x2
0
1
2
3
x1
0
0.08
0.06
0.04
0.00
1
0.07
0.13...

Consider the following bivariate distribution p(x, y) of two
discrete random variables X and Y.
Y\X
-2
-1
0
1
2
0
0.01
0.02
0.03
0.10
0.10
1
0.05
0.10
0.05
0.07
0.20
2
0.10
0.05
0.03
0.05
0.04
a) Compute the marginal distributions p(x) and p(y)
b) The conditional distributions P(X = x | Y = 1)
c) Are these random variables independent?
d) Find E[XY]
e) Find Cov(X, Y) and Corr(X, Y)

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