Question

A fair die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a 5 and a 4. Find (a) E X, (b) E[X|Y = 1] and (c) E[X|Y = 4].

Answer #1

A fair die is successively rolled. Let X and Y denote,
respectively, the number of rolls necessary to obtain a 5 and a 4.
Find (a) EX, (b) E[X|Y =1] and (c) E[X|Y=4].

a fair die was rolled repeatedly.
a) Let X denote the number of rolls until you get at least 3
different results. Find E(X) without calculating the distribution
of X.
b) Let S denote the number of rolls until you get a repeated
result. Find E(S).

A fair die is rolled repeatedly. Let X be the random variable
for the number of times a fair die is rolled before a six appears.
Find E[X].

Alice rolls a six faced fair die twice, and she obtains two
numbers that we denote with X and Y , respectively. Fair die means
that, in each roll, the six outcomes are equally likely. Let A be
the event that X = 4, B be the event that X + Y = 6, and C be the
event that Y = 5. (a) Write the sample space S for this experiment.
(b) Are A and B independent? (c) Are...

8 Roll a fair (standard) die until a 6 is obtained and let Y be
the total number of rolls until a 6 is obtained. Also, let X the
number of 1s obtained before a 6 is rolled.
(a) Find E(Y).
(b) Argue that E(X | Y = y) = 1/5 (y − 1). [Hint: The word
“Binomial” should be in your answer.]
(c) Find E(X).

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.

A die is rolled six times.
(a) Let X be the number the die obtained on the first roll. Find
the mean and variance of X.
(b) Let Y be the sum of the numbers obtained from the six rolls.
Find the mean and the variance of Y

A fair die is rolled once. Let A = the die shows an odd
number. Let B = the die shows a number
greater than 4.
(a) Find A ∪ B.
(b) Find A ∩ B.
(c) Find P(A ∪ B)

A fair die is continually rolled until an even number has
appeared on 10 distinct rolls. Let Xi denote the number
of rolls that land on side i. Determine
(a) E[X1]
(b) E[X2]
(c) the probability mass function of X1
(d) the probability mass function of X2
Same Questions are on Q&A but I want to know why X1 is
geometric. I know that geometric distribution means that the number
of trials for first success.
Thanks

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...

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