Question

**Two identical fair 6-sided dice are rolled
simultaneously. Each die that shows a number less than or equal to
4 is rolled once again. Let X be the number of dice that show a
number less than or equal to 4 on the first roll, and let Y be the
total number of dice that show a number greater than 4 at the
end.**

**(a) Find the joint PMF of X and Y . (Show your final
answer in a table.)**

**(b) Find the marginal PMF of X.**

**(c) Compute the mean and variance of X.**

**(d) Find the conditional PMF of Y given X =
1.**

**(e) Compute the conditional mean and variance of Y given
X = 1.**

Answer #1

a) The joint PDF for X, Y is obtained here as:

X = 0 | X = 1 | X = 2 | |

Y = 0 | 0 | 0 | (1/3)^{4} = 1/81 |

Y = 1 | 0 | (2c1)*(2/3)*(1/3)*(1/3) = 4/27 | (1/3)^{3}(2/3)*2 = 4/81 |

Y = 2 | (2/3)^{2} = 4/9 |
(2c1)*(2/3)*(1/3)*(2/3) = 8/27 | (1/3)^{2}*(2/3)^{2} = 4/81 |

b) The marginal PDF from above joint PDF now could be obtained as:

P(X = 0) = 4/9

P(X = 1) = 12/27 = 4/9

P(X = 2) = 1/9

P(Y = 0) = 1/81

P(Y = 1) = 16/81

P(Y = 2) = 64/81

c) The mean of X is computed as:

E(X) = 1*4/9 + 2/9 = 6/9 = **2/3**

E(X^{2}) = 1*4/9 + 2^{2}/9 =
**8/9**

Var(X) = E(X^{2}) - [E(X)]^{2} = (8/9) -
(2/3)^{2} = **4/9**

d) P(Y = 0 | X = 1) = **0**

P(Y = 1 | X = 1) = 4/12 = **1/3**

P(Y = 2 | X = 2) = **2/3**

e) E(Y | X = 1) = 1*(1/3) + 2*(2/3) = **5/3**

E(Y^{2} | X = 1) = 1*(1/3) + 2^{2}*(2/3) = 3

Var(Y | X = 1) = 3 - (5/3)^{2} = **2/9**

A fair six-sided die is rolled 10 independent times. Let X be
the number of ones and Y the number of twos.
(a) (3 pts) What is the joint pmf of X and Y?
(b) (3 pts) Find the conditional pmf of X, given Y = y.
(c) (3 pts) Given that X = 3, how is Y distributed
conditionally?
(d) (3 pts) Determine E(Y |X = 3).
(e) (3 pts) Compute E(X2 − 4XY + Y2).

4 fair 10-sided dice are rolled.
(a)
Find the conditional probability that at least one die lands on
3 given that all 4 dice land on different numbers.
(b)
True or False: If X is the sum of the 4 numbers from
one roll, and Y is the maximum of the 4 numbers from one
roll, then X and Y are independent random
variables.

Consider rolling two fair six-sided dice.
a) Given that the roll resulted in sum of 8, find the
conditional probability that first die roll is 6.
b) Given that the roll resulted in sum of 4 or less, find the
conditional probability that doubles are rolled.
c) Given that the two dice land on different numbers, find the
conditional probability that at least one die is a 6.

5 fair 8-sided dice are rolled.
(a)
[3 marks] Find the conditional probability that at least one
die lands on 3 given that all 5 dice land on different
numbers.
(b)
[2 marks] True or False: If X is the minimum of the 5
numbers from one roll, and Y is the sum of the 5 numbers
from one roll, then X and Y are independent
random variables.

Two fair six-sided dice are rolled once. Let (X, Y) denote the
pair of outcomes of the two rolls.
a) Find the probability that the two rolls result in the same
outcomes.
b) Find the probability that the face of at least one of the
dice is 4.
c) Find the probability that the sum of the dice is greater than
6.
d) Given that X less than or equal to 4 find the probability
that Y > X.

PROBLEM #2
Suppose you play a game in which a fair 6 sided die is rolled
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facing upward) is less than or equal to 4, you are paid as many
dollars as the number you have rolled. Otherwise, you lose as many
dollars as the number you have rolled.
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5 fair 8-sided dice are rolled.
(a)
Find the conditional probability that at least one die lands on
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Consider an experiment where we roll 7 fair 6-sided dice
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independent from each other).
(a) What is the probability that exactly 3 of the dice are greater
than or equal to 5?
(b) Suppose now that each of the 7 6-sided dice are weighted the
same such that the probability of
rolling a 6 is 0.5, and every other side that is not a 6 has equal
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Imagine rolling two fair 6 sided dice. the number rolled on the
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We roll two fair 6-sided dice, A and B. Each one of the 36
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a. Find the probability that dice A is larger than dice B.
b. Given that the roll resulted in a sum of 5 or less, ﬁnd the
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c. Given that the two dice land on different numbers, ﬁnd the
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