Question

Let X be the number of 1’s and Y the number of 2’s that occur in 4 rolls of a fair die. Compute Cov(X, Y ).

Answer #1

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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 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].

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).

roll a fair die 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).

I roll a fair die until I get my first ace. Let X be the number
of rolls I need.
You roll a fair die until you get your first ace. Let Y be the
number of rolls you need.
(a) Find P( X+Y = 8)
HINT: Suppose you and I roll the same die, with me going first.
In how many ways can it happen that X+Y = 8, and what is the
probability of each of those ways?...

Let X equal the outcome (1, 2 , 3 or 4) when a fair four-sided
die is rolled; let Y equal the outcome (1, 2, 3, 4, 5 or 6) when a
fair six-sided die is rolled. Let W=X+Y.
a. What is the pdf of W?
b What is E(W)?

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

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).

1) A 10-sided die is rolled infinitely many times. Let X be the
number of rolls up to and including the first roll that comes up 2.
What is Var(X)?
Answer: 90.0
2) A 14-sided die is rolled infinitely many times. Let X be the
sum of the first 75 rolls. What is Var(X)?
Answer: 1218.75
3) A 17-sided die is rolled infinitely many times. Let X be the
average of the first 61 die rolls. What is Var(X)?
Answer:...

Let A1, ... ,20 be independent events each with probability 1/2.
Let X be the number of events among the first 10 which occur and
let Y be the number of events among the last 10 which occur. Find
the conditional probability that X = 5, given that X + Y = 12.

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