Question

Let X1,X2 be two independent rolls of a standard 6-sided die, and let Y = X1+X2.

a) Determine the distribution of Y explicitly. Give your answer in table format.

b) Calculate E(Y ) using the distribution you found in part a).

c) Explain your answer to part b) in terms of E(X1) and E(X2).

That's full problem, Thank you!!

Answer #1

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...

Suppose you roll two twenty-five-sided dice. Let X1, X2 the
outcomes of the rolls of these two fair dice which can be viewed as
a random sample of size 2 from a uniform distribution on
integers.
a) What is population from which these random samples are drawn?
Find the mean (µ) and variance of this population (σ 2 )? Show your
calculations and results.
b) List all possible samples and calculate the value of the
sample mean ¯(X) and variance...

Suppose X1 and X2 are independent expon(λ) random variables. Let
Y = min(X1, X2) and Z = max(X1, X2).
(a) Show that Y ∼ expon(2λ)
(b) Find E(Y ) and E(Z).
(c) Find the conditional density fZ|Y (z|y).
(d) FindP(Z>2Y).

Let X1 and X2 be independent random variables such that X1 ∼ P
oisson(λ1) and X2 ∼ P oisson(λ2). Find the distribution of Y = X1 +
X2.s

Let X1, X2, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is
the middle of the three values). Find the conditional CDF of X1,
given the event Y = 1/2. Under this conditional distribution, is X1
continuous? Discrete?

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]

You are given that X1 and X2 are two independent and identically
distributed random variables with a Poisson distribution with mean
2. Let Y = max{X1, X2}. Find P(Y = 1).

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).

Let Y be the liner combination of the independent random
variables X1 and X2 where Y = X1 -2X2
suppose X1 is normally distributed with mean 1 and standard
devation 2
also suppose the X2 is normally distributed with mean 0 also
standard devation 1
find P(Y>=1) ?

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).

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