Suppose two fair sided die with sides labeled 1,2,3,4,5,6 are
tossed independently.
Let X = the minimum of the value from each die.
a. What is the probability mass function(pmf) of X?
b. Find the mean E[X] and variance V (X).
c. Write the cumulative distribution function (cdf) of X in a
tabular form.
d. Write F(x) the cdf of X as a step function and give a rough
sketch for this function.
A die is rolled and, independently, a coin is tossed. Let X be
the value of the die if the coin is H and minus the value of the
die if the coin is T.
(a) Calculate and plot the PMF of X.
(b) Calculate E [X] and var(X)
(c) Calculate and plot the PMF of X 2 − 2X.
2. A drunk walks down a street. Assume he starts at block 0.
Every 10 minutes, he moves north a...
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, and a fair die is rolled. Let H be the
event that the coin lands on heads, and let S be the event that the
die lands on six. Find P(H or S).
Two fair six-sided dice are tossed independently. Let M
= the maximum of the two tosses (so M(1,5) = 5,
M(3,3) = 3, etc.).
(a) What is the pmf of M? [Hint: First
determine p(1), then p(2), and so on.] (Enter
your answers as fractions.)
m
1
2
3
4
5
6
p(m)
(b) Determine the cdf of M. (Enter your answers as
fractions.)F(m) =
m < 1
1 ≤ m <...
A fair die is tossed 120 times. A success is getting a 2 or a 5
landing face up.
(a) Find the probability of getting exactly 40 successes.
(b) Does the normal approximation to the binomial apply? If so,
use it to approximate the probability of exactly 40 successes.
7. (Sec. 3.2) Two fair six-sided dice are tossed independently.
Let M = the minimum of the two tosses. For example, M(2, 5) = 2,
M(4, 4) = 4, etc.
(a) What is the PMF of M? [Hint: just work out each probability
individually by counting the number of outcomes which result in a
specific value for M, i.e. find p(1), then p(2), and so on up to
p(6)].
(b) Determine the CDF of M. (
c) Graph the CDF...
Two fair coins and a fair die are tossed. Find the sample space
of the
experiment (10 pts); Find the probabilities of the following
events:
A- ”the die shows 2 or 3” (5 pts);
B- ”one of the coins is head, the other - tail, and the die shows
an odd number” (5
pts).
Are the events A and B independent? (5 pts).
Give proofs.
Two fair dice are tossed. Let A be the maximum of the two
numbers and let B be the absolute difference between the two
numbers. Find the joint probability of A and B. Are A and B
independent? How do you know?
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).