Question

- Consider the probability distribution of the discrete random vector [X,Y] where X represents the number of orders for chickens in August at neighboring supermarket and Y represents the number of orders in September. The Joint distribution is showing the following table.

Y |
51 |
52 |
53 |
54 |
55 |

51 |
0.06 |
0.05 |
0.05 |
0.01 |
0.01 |

52 |
0.07 |
0.05 |
0.01 |
0.01 |
0.01 |

53 |
0.05 |
0.10 |
0.10 |
0.05 |
0.05 |

54 |
0.05 |
0.02 |
0.01 |
0.01 |
0.03 |

55 |
0.05 |
0.06 |
0.05 |
0.01 |
0.03 |

- Find the probability that X ≥ 53 and Y ≥ 53 (3Marks)
- Find the marginal distribution of X? (2marks)
- Find the marginal distribution of X?(2marks)
- Find the expected sales for September i.e. E(Y). (3 Mark)
- Find P(Y≥53 | X =55)
- Calculate the correlation coefficient of X and Y (6Marks)

Answer #1

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)

Airlines sometimes overbook flights. Suppose that for a plane
with 50 seats, 55 passengers have tickets. Define the random
variable Y as the number of ticketed passengers who
actually show up for the flight. The probability mass function of
Y appears in the accompanying table.
y
45
46
47
48
49
50
51
52
53
54
55
p(y)
0.05
0.10
0.12
0.14
0.25
0.17
0.05
0.02
0.06
0.01
0.03
Calculate V(Y) and
σY. (Round your variance to
four decimal places...

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)

Problem 3. Let x be a discrete random variable with the
probability distribution given in the following table:
x = 50 100 150 200 250 300 350
p(x) = 0.05 0.10 0.25 0.15 0.15 0.20 0.10
(i) Find µ, σ 2 , and σ.
(ii) Construct a probability histogram for p(x).
(iii) What is the probability that x will fall in the interval
[µ − σ, µ + σ]?

In the accompanying table, the random variable x represents the
number of televisions in a household in a certain country.
Determine whether or not the table is a probability distribution.
If it is a probability distribution, find its mean and standard
deviation.
x P(x)
0 0.03
1 0.13
2 0.32
3 0.25
4 0.17
5 0.10
If the table is a probability distribution, what is its mean?
Select the correct choice below and fill in any answer boxes within
your...

The probability distribution of a couple of random variables (X,
Y) is given by :
X/Y
0
1
2
-1
a
2a
a
0
0
a
a
1
3a
0
a
1) Find "a"
2) Find the marginal distribution of X and Y
3) Are variables X and Y independent?
4) Calculate V(2X+3Y) and Cov(2X,5Y)

X and Y are continuous random variables. Their joint probability
distribution function is :
f(x,y) = 1/5(y+2) , 0 < y < 1, y-1 < x < y +1
= 0, otherwise
a) Find marginal density of Y, fy(y)
b) Calculate E[X | Y = 0]

Create the probability distribution for a random variable X that
represents the following:The number of girls in a 3 child
family.

If the joint probability distribution of X and Y is given
by:
f (x, y) = 3k (x + y), for x = 0, 1, 2, 3; y = 0, 1, 2.
a) .- Find the constant k.
b) .- Using the table of the joint distribution and the
marginal distributions, determine if variable X and variable Y are
independent.

The joint probability distribution of two random variables X and
Y is given in the following table
X Y →
↓
0
1
2
3
f(x)
2
1/12
1/12
1/12
1/12
3
1/12
1/6
1/12
0
4
1/12
1/12
0
1/6
f(y)
a) Find the marginal density of X and the marginal density of Y.
(add them to the above table)
b) Are X and Y independent?
c) Compute the P{Y>1| X>2}
d) Compute the expected value of X.
e)...

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