a) A coin is tossed 4 times. Let X be the number of Heads on the
first 3 tosses and Y be the number of Heads on the last three
tossed. Find the joint probabilities pij = P(X = i, Y = j) for all
relevant i and j. Find the marginal probabilities pi+ and p+j for
all relevant i and j. b) Find the value of A that would make the
function Af(x, y) a PDF. Where f(x, y)...
A coin is tossed 6 times. Let X be the number of Heads in the
resulting combination. Calculate the second moment of X.
(A).Calculate the second moment of X
(B). Find Var(X)
A coin is tossed twice. Let Z denote the number of heads on the
first toss and W the total number of heads on the 2 tosses. If the
coin is unbalanced and a head has a 40% chance of occurring, find
the correlation between W and Z.
A coin is tossed twice. Let Z denote the number of heads on the
first toss and W the total number of heads on the 2 tosses. If the
coin is unbalanced and a head has a 40% chance of occurring, find
the correlation between W and Z
Flip a fair coin 100 times. Let X equal the number of heads in
the first 65 flips. Let Y equal the number of heads in the
remaining 35 flips.
(a) Find PX (x) and PY (y).
(b) In a couple of sentences, explain whether X and Y are or are
not independent?
(c) Find PX,Y (x, y).
A fair coin is tossed four times. Let X denote the number of
heads occurring and let Y denote the longest string of heads
occurring. (i) determine the joint distribution of X and Y (ii)
Find Cov(X,Y) and ρ(X,Y).
Toss a coin 5 times. Let X denote the number of tails appeared.
a. Write down the probability mass function of X. b. Write down the
cumulative distribution function of X. c. Graph the cumulative
distribution function of X. d. Find the expectation of E[X] e. Find
the variance Var[X]
Toss a fair coin repeatedly. Let N1
be the number of tosses required to obtain heads followed
immediately by tails. Let N2 be the number of tosses
required to obtain two heads in a row.
(A) Should N1 and N2
have the same expected value? If not, which expected value should
be larger? Explain your answers.
(B) Find the probability mass function of
N1.
(C) Find the expected value of N1.
(D) Find the probability mass function of
N2.
(E)...
Q7 A fair coin is tossed three times independently: let X denote
the number of heads on the first toss (i.e., X = 1 if the first
toss is a head; otherwise X = 0) and Y denote the total number of
heads.
Hint: first figure out the possible values of X and Y , then
complete the table cell by cell.
Marginalize the joint probability mass function of X and Y in
the previous qusetion to get marginal PMF’s.
A fair coin is tossed r times. Let Y be the number of heads in
these r tosses. Assuming Y=y, we generate a Poisson random variable
X with mean y. Find the variance of X. (Answer should be based on
r).