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Math/Stat 425, Solutions to Quiz #7
Problem # 1. Let X and Y be independent random variables uniformly distributed
on the unit interval [0, 1]. Compute P [Y < X + 0.5].
Solution: The density function for the uniform distribution is
fX,Y (x, y ) = 1 for 0
Math/Stat 425, Solutions to Quiz #6
Problem # 2. Let X be a continuous random variable with the probability density
function
1
x if 0 < x < 2
2
f (x) = 0 otherwise
f (x) =
Find the expected value E [X ].
Solution: By denition
Answer:
2
xf (x)dx =
E [X ] =
Math/Stat 425, Solutions to Quiz #6
Problem # 1. Let X be a continuous random variable with the probability density
function
x
2
if 0 < x < 2
f (x) =
0
otherwise
Let Y = X 2 . Find the cumulative distribution function of Y .
(That is, give FY (t), for t 0
Math/Stat 425, Solutions to Quiz #5
Problem # 2. Let X be the number of chips in a randomly selected chocolate chip
cookie. Assuming that X is a Poisson random variable with E [X ] = 5, compute the
probability that a randomly selected cookie has at least
Math/Stat 425, Solutions to Quiz #5
Problem # 1. A fair die is rolled 10 times. What is the probability that a 6 comes
up at least three times? [Answer can be expressed using factorials, sums. State what
distribution you use.]
Solution: Consider the rando
Math/Stat 425, Solutions to Quiz #4
Problem # 2. Six people check their coats, and then coats are handed back at random
so that each person gets precisely one coat. Let X be the number of people who get
their own coat. Find the expectation E (X ) of X .
H
Math/Stat 425, Solutions to Quiz #4
Problem # 1. A box contains 3 red marbles and 2 green marbles. Suppose that one
picks 3 marbles randomly without replacement from the box. Let X count the number of
red marbles minus the number of green marbles among th
Math/Stat 425: Solutions to Quiz 3
Problem # 2. A fair coin is tossed three times. What is the probability
that the coin shows heads on the second toss, given that it showed heads
exactly twice?
Solution: The sample space is
S = cfw_HHH, HHT, HT H, T HH,
Math/Stat 425: Solutions to Quiz 3
Problem # 1. There is a box containing 9 red balls and 18 blue balls.
Suppose that one draws 6 balls at random from the box (without replacement). What is the probability that this draw consists of exactly
2 red balls an
Math/Stat 425, Solutions to Quiz #2
Problem # 2. A student must choose exactly two of three electives: Art, Business, or
Communications. The student chooses Art with probability 5/8, Business with probability 5/8 and Art and Business together with probabi
Math/Stat 425, Solutions to Quiz #2
Problem # 1. For two events A and B in a probability space S, we have
3
P (A) = P (B ) = .
4
(a) Suppose we know that P (A B ) = 5 . What is P (AB )?
6
(b) Suppose that we dont know P (A B ). What can be the minimum pos
Math/Stat 425, Solutions to Quiz #1
Problem # 2. Consider a group of 9 dierent people. Two committees are to be
formed, one of four people and one of three people. (No person can serve on both committees.) In how many ways can the committees be formed. (P
Math/Stat 425, Solutions to Quiz #1
Problem # 1. We are given a 5 5 square grid with vertices A = (0, 0), B = (3, 2),
and C = (5, 5) marked. We want to count paths between these points that move only
right or upwards along the grid.
(a) How many dierent s
Solutions - Homework #3
1
1.1
Chapter 2
Problems
1. Problem 17: If 8 castles (rooks) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or
Math/Stat 425, Solutions to Quiz #7
Problem # 2. Let X and Y be independent random variables, having the joint probability density function
f (x, y ) = 4xy
if 0 < x < 1, and 0 < y < 1.
and f (x, y ) = 0 otherwise. Compute the probabilityP [X + Y < 1].
Sol