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 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).
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).
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 coin is tossed three times. X is the random variable for the
number of heads occurring.
a) Construct the probability distribution for the random
variable X, the number of head occurring. b) Find P(x2). c) Find
P(x1). d) Find the mean and the standard deviation of the
probability distribution for the random variable X, the number of
heads occurring.
A coin with probability p>0 of turning up heads is tossed 4
times. Let X be the number of times heads are tossed.
(a) Find the probability function of X in terms of p.
(b) The result above can be extended to the case of n
independent tosses (that is, for a generic number of tosses), and
the probability function in this case receives a very specific
name. Find the name of this particular probability function.
Notice that the probability...
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 ...
Suppose a coin is tossed 100 times and the number of heads are
recorded. We want to test whether the coin is fair. Again, a coin
is called fair if there is a fifty-fifty chance that the outcome is
a head or a tail. We reject the null hypothesis if the number of
heads is larger than 55 or smaller than 45.
Write your H_0 and H_A in terms of the probability of heads, say
p.
Find the Type I...
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).