Question

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].

Answer #1

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).

Consider the following three random variables:
a) a coin is repeatedly flipped
Let X= number of tails before the first head
b) a box contains 10 red and 6 white chips. 5 chips are drawn at
rabdom from the box.
Let Y= the number of red chips drawn.
c) A weighted die is rolled 12 times.
Let W= the number of times a "4" is rolled
in each case, decide whether the random variable is
binomial.

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

You roll a six-sided die repeatedly until you roll a one. Let X
be the random number of times you roll the dice. Find the following
expectation:
E[(1/2)^X]

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 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 is rolled repeatedly. Find the expected number of
rolls until all 6 faces appear.

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 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)

Consider an experiment where a fair die is rolled repeatedly
until the first time a 3 is observed.
i) What is the sample space for this experiment? What is the
probability that the die turns up a 3 after i rolls?
ii) What is the expected number of times we roll the die?
iii) Let E be the event that the first time a 3 turns up is after
an even number of rolls. What set of outcomes belong to...

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