Question
In: Statistics and Probability
A fair coin is tossed four times. Let X denote the number of heads occurring and...
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).
Solutions
orchestra answered 2 years agoRelated Solutions
Q7 A fair coin is tossed three times independently: let X denote the number of heads...
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...
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).
Flip a fair coin 4 times. Let ? and ? denote the number of heads and...
Flip a fair coin 4 times. Let ? and ? denote the number of heads
and tails correspondingly.
(a) What is the distribution of ?? What is the distribution of ?
?
(b) Find the joint PMF. Are ? and ? independent?
(c) Calculate ?(? ?) and ?(X≠?)(d) Calculate C??(?, ? ) and
C???(?, ? )
a) A coin is tossed 4 times. Let X be the number of Heads on the...
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...
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)
An unbiased coin is tossed four times. Let the random variable X denote the greatest number...
An unbiased coin is tossed four times. Let the random variable X
denote the greatest number of successive heads occurring in the
four tosses (e.g. if HTHH occurs, then X = 2, but if TTHT occurs,
then X = 1).
Derive E(X) and Var(X).
(ii) The random variable Y is the number of heads occurring in
the four tosses. Find Cov(X,Y).
If a fair coin is tossed 25 times, the probability distribution for the number of heads,...
If a fair coin is tossed 25 times, the probability distribution
for the number of heads, X, is given below. Find the mean and the
standard deviation of the probability distribution using Excel
Enter the mean and round the standard deviation to two decimal
places.
x P(x)
0 0
1 0
2 0
3 0.0001
4 0.0004
5 0.0016
6 0.0053
7 0.0143
8 0.0322
9 0.0609
10 0.0974
11 0.1328
12 0.155
13 0.155
14 0.1328
15 0.0974
16 ...
A coin is tossed twice. Let Z denote the number of heads on the first toss...
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...
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...
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).
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