Question

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: 0.3934

**I know the answers but don't know how to get them.
Please show work**

Answer #1

1)

here this is geometric distribution with paramter p=1/10 (cause each number has equal probability)

Var(X) =(1-p)/p^{2} =(1-1/10)/(1/10)^{2}
=100*9/10=90

2)

expected value on a single roll E(X)=(1+2+3+4+5+6+7+8+9+10+11+12+13+14)/14 =7.5

and
E(X^{2})=(1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2}+11^{2}+12^{2}+13^{2}+14^{2})/14=72.5

therefore variance of single roll =
E(X^{2})-(E(X))^{2} =16.25

for 75 rolls variance of sum =16.25*(75)=1218.75

3)

as above expected value E(X)=9

E(X^{2})=105

Var(X)=24

therfore variance of average of 61 die rolls=24/61=0.3934

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 6-sided die rolled twice. Let E be the event "the first roll
is a 1" and F the event "the second roll is a 1".
Find the probability of showing a 1 on both rolls. Write your
answer as a reduced fraction.

You have a 4 sided die and 10 sided die that are rolled 50
times. What is the theoretical probability that the small die is
odd, the sum of the numbers is 5, the same # appears on both, the #
on the larger die is > than the # on the smaller die.

Assume that a fair
six-sided die is rolled 9 times, and the roll is called a success
if the result is in {1,2}{1,2}.
What is the probability that there are exactly 4 successes or
exactly 4 failures in the 9 rolls?

A 10 sided die with sides numbered 0 through 9 is rolled 5
times. Let X be the number of zeros rolled.
(a) Find the probability that exactly 2 zeros are rolled.
(b) Find the probability that no less than 4 zeros are
rolled.
(c) Find the mean and the variance of X.

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 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 4-sided die is rolled 7 times.
(a)
Find the probability that the side 1 comes up exactly 3
times.
(b)
Find the probability that there is at least one side that comes
up exactly 3 times.

You roll a fair 6 sided die 5 times. Let Xi be the
number of times an i was rolled for i = 1, 2, . . . , 6.
(a) What is E[X1]?
(b) What is Cov(X1, X2)?
(c) Given that X1 = 2, what is the probability the
first roll is a 1?
(d) Given that X1 = 2, what is the conditional probability mass
function of, pX2|X1 (x2|2), of
X2?
(e) What is E[X2|X1]

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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