Question

Roll a pair of fair dice. Let X be the number of ones in the outcome and let Y be the number of twos in the outcome. Find E[XY].

Answer #1

from above:

x | y | P(x,y) | xy*P(X,Y) |

0 | 0 | 4/9 | 0 |

0 | 1 | 2/9 | 0 |

1 | 0 | 2/9 | 0 |

1 | 1 | 1/18 | 1/18 |

2 | 0 | 1/36 | 0 |

0 | 2 | 1/36 | 0 |

1 | 1/18 |

from above:

**E(XY)=1/18**

Roll a pair of fair dice. Let X be the number of ones in the
outcome and let Y be the number of twos in the outcome. Are X and Y
independent?

You roll a pair of fair dice repeatedly. Let X denote the number
of rolls until you get two consecutive sums of 8(roll two 8 in a
row). Find E[X]

Roll a die and let its outcome be the random variable X. Let Y
be the random variable of “sum of X many dice rolled”. So, if X is
3, then we roll 3 dice and add the faces together to find Y .
(a) Are X and Y independent? Explain.
(b) Compute E[Y]

Roll a die and let its outcome be the random variable X. Let Y
be the random variable of “sum of X many dice rolled”. So, if X is
3, then we roll 3 dice and add the faces together to find Y .
(a) Are X and Y independent? Explain.
(b) Compute E[Y]

Suppose we roll a pair of fair dice, let A=the numbers I rolled
add up to exactly 8, and let B=the numbers I rolled multiply to an
even number. Find P(Bc|Ac)

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).

A pair of fair dice is tossed and the up face values are
observed. Define X to be the number of twos observed. Find the mean
of X.
Round your answer to the nearest thousandths.

A pair of fair dice is tossed and the up face values are
observed. Define X to be the number of ones observed. Find P(X =
0).

Roll a fair four-sided die twice. Let X be the sum of the two
rolls, and let Y be the larger of the two rolls (or the common
value if a tie).
a) Find E(X|Y = 4)
b) Find the distribution of the random variable E(X|Y )
c) Find E(E(X|Y )). What does this represent?
d) Find E(XY |Y = 4)
e) Find the distribution of the random variable E(XY |Y )
f) Explain why E(XY |Y ) = Y...

roll a pair of fair dice and at least one of the dice is a
six.
without any further information to go on, what is the
conditional probability that the sum of the dice is 8? Is the event
that “sum is 8” independent from the event that “at least one of
the dice is a six”?

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