STAT346, section 002 Homework 3 solutions
2-17-15 correction to 4b. The previous answer was correct, but the notation was not. Ive also
added a bit of explanation to the answer.
1. A box contains four bags with the following contents.
One bag has twelve
Midterm Exam 1
STAT 346, Spring 2011
Instructions: You cannot use any books or notes. You can use a calculator, but not a computer. You are
not allowed to communicate with anyone (verbally, in writing, or electronically), except for me, during the
exam pe
Homework 2 solutions
1. The area of the United States composed of the six states Maine, Massachusetts, New Hampshire, Vermont, Rhode Island, and Connecticut is called New England. If for a randomly
selected person from the US, the probability they were bo
1. Consider a series of contests between two teams (A and B) with the winner of the series the first team to win three games. The results can be described by the following tree diagram. The letters describe the winners of each game. The red circles indica
Solutions for HW 3
STAT 346, Spring 2013
1) Letting A be the event that at least one card is black, and E be the event that exactly one card is black,
for the desired probability we have
P (E |A) =
which is equal to
P (E A)
P (E )
=
=
P (A)
1 P (AC )
26
1
STAT346, fall 2015, Quiz 2 solutions
1. Let A and B be events. You may use either logic or set theory notation to answer either (a)
or (b).
(a) Write P(A or B) as a function of P(B) and P(A and B c ).
(b) Write P(A B) as a function of P(B) and P(A B c ).
STAT346 2015 Midterm solutions
PLEASE LET ME KNOW IF YOU FIND ANY MISTAKES/TYPOS IN THIS.
Correction made to questions 2(c) 6:30pm 04/06/15
1. For each of the following inequalities, put a check in the box if it indicates that X and
Y are not independent
STAT346, spring 2015, Quiz 3 solutions
1. P(X = x|Y = y, Z = z) =
P(Y = y|X = x, Z = z)C
P(Y = y|Z = z)
C = P(X = x|Z = z)
This is just Bayes rule. Without the conditioning on Z = z we have
P(X = x|Y = y) =
P(Y = y|X = x)P(X = x)
P(Y = y)
To use this form
STAT346, fall 2015, Quiz 1 solutions
(1) P(Ac ) = 1 P(A and B) P(A and B c )
P(Ac ) = 1 P(A B) P(A B c )
always true
P(Ac ) = 1 P(A)
= 1 P(A and B) + P(A and B c )
= 1 P(A and B) P(A and B c )
(2) P(A or B) = P(A and B) + P(A and B c ) + P (Ac and B)
P(A
STAT346, section 002 Homework 8 Solutions
1. Let X be a discrete random variable with support cfw_1, . . . , n and density function
P(X = x) =
1
n
for x = 1, . . . , n,
(1)
where n is an integer. The distribution of a random variable with this density fun
STAT346, section 002 Homework 10 solutions
When it exists, the moment generating function for a variable X is dened as
Mx (t) = E etX = E [exp(tX)] .
Moment generating function is commonly abbreviated as mgf. Many of the commonly used distributions have m
STAT346, section 002 Homework 11 solutions
Theorem 1
Let Y1 , Y2 , . . . , Yk be independent random variables, where for i = 1, . . . , k, Yi Binomial(ni , p).
Dene
k
W =
Yi .
i=1
Then W Binomial(n1 + n2 + + nk , p).
Theorem 2
Let Y be a normally distribu
STAT346, section 002 Homework 9 solutions
The density function for the Gamma distribution is
f (x) =
1
x1 ex/ for 0 x < ,
(a)
, > 0,
(1)
() is called the Gamma function. It is dened for all real numbers. For any a (, )
(a)
ua1 eu du,
0
where means is de
STAT346, section 002 Homework 7 solutions
1. Let X be a discrete random variable with support cfw_0, 4, 9, P(X = 0) = 1/6, and
P(X = 4) = 1/2.
(This is similar, but not identical to the question we did in class.)
(a) Draw a graph of the density function.
STAT346, section 002 Homework 6 (There are 6 questions, which start on page 3.)
Due at the beginning of class on March 26, 2015
There are probably typos in this. Please let me know if you nd any.
Any random variable with support cfw_0, 1 is called a Berno
STAT346, section 002 Homework 4 solutions
1. I have underlined the key words that indicate you are being given a conditional probability.
Suppose that 50% of the population are male, 5% of the men are color-blind, and 0.25%
of the women are color-blind. I
STAT346, section 002 Homework 5 solutions
This question is from A First Course in Probability, Ross, and is posted on blackboard.
However, I use dierent (and better) notation.
A plane is missing, and it is presumed that it is equally likely to have gone d
STAT346, section 002 Homework 1 solutions
1. Tenants in a large apartment complex are allowed to own dogs, cats, and rabbits, but
are not allowed to own more than two pets. Write down the sample space for the pets
owned by the tenants in a randomly select
Additional Important Problems
1) If X has mgf
1
2 et
(t < log 2),
2xey ,
0,
MX (t) =
x > 0, y > x2 ,
otherwise,
give the value of E (X ).
answer: 1
2) Using the joint pdf,
f (x, y ) =
obtain each of the following:
(a) the marginal pdf of X ,
(b) the margi
STAT346, fall 2015, Quiz 4 Solutions
1. Let X be a discrete random variable with distribution function
t<2
0
1
13 2 t < 4
4 4t<6
F (t) =
13
9
13 6 t < 8
1
t 8.
(a) What is the support?
cfw_2, 4, 6, 8
(b) Find the density function.
P(X = 2) = P(X 2) =
1
STAT346, fall 2015, Quiz 5 solutions
1. For each joint distribution function below, indicate if X and Y are independent or not independent, and show and/or explain how you know.
(a) f (x, y) = xex(y+1) ,
0 x < ,
0y<
X and Y are not independent because the
Homework #1 STAT346 Section 003
Question 1: An experiment of flipping a coin is performed until both heads and tails are seen
for at least once. Let E be the event that the experiment stops after the coin has been flipped
at most 4 times. Let F be the eve
Solutions for HW 4
STAT 346, Spring 2013
1)
(a) The desired pmf is
0.3,
0.5,
pX (x) =
0.2,
0,
x = 5,
x = 1,
x = 1,
otherwise.
(b) We have that
P (X > 3) = 1 P (X 3) = 1 FX (3) = 1 0.7 = 0.3.
Alternatively, one can simply note that X has only one possibl
Solutions for HW 5
STAT 346, Spring 2013
1) We need
4 = Var(Y ) = Var(aX + b) = a2 Var(X ) = a2 /4,
which implies that a2 = 16, which means that we must have a = 4 (due to the requirement that a be
positive). We also need
10 = E (Y ) = E (aX + b) = aE (X
Solutions for HW 6
STAT 346, Spring 2012
1) Letting X be the number of times a black ball is obtain n 9 trials, X is a binomial (9, 1/3) random
variable. For parts (d) through (g), Y is a geometric random variable, and for part (h), V is a negative
binomi
Solutions for HW 7
STAT 346, Spring 2013
1)
(a) The support of X is (2, ) and so FX (x) = 0 for x 2. For x > 2,
x
FX (x) = P (X x) =
2
24t4 dt = 8t3 |x = 1 8x3 .
2
Altogether, we have
1
0,
FX (x) =
8
x3 ,
x > 2,
x 2.
(b) We have P (X > 4) = 1 P (X 4) = 1
Solutions for HW 8
STAT 346, Spring 2013
1)
(a) One way to obtain the desired pdf is to make use of Theorem 6.1 on p. 243 of the text. Letting
d
d
Y = h(X ) = X 3 , we have h1 (x) = x1/3 and dx h1 (x) = dx x1/3 = x2/3 /3. It follows from the
theorem that
Solutions for HW 9
STAT 346, Spring 2013
1) For t > 0 we have
P (T > t) =
t
(v + 1)2 dv = (v + 1)1 | = (t + 1)1 .
t
So, for t > 0, the desired hazard rate function is
fT (t)/P (T > t) = [1/(t + 1)2 ]/[1/(t + 1)] = 1/(t + 1).
2)
(a) We have
1 = pX,Y (1, 1)
HW 10
STAT 346, Spring 2013
1) Below Ill list all of the possible outcomes, their probabilities, and the corresponding values of Y and Z .
outcome
000
100
010
001
110
101
011
111
prob.
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
y
0
1
1
1
2
2
2
3
z
0
-1
0
0
-2
0
1
0
HW 11
STAT 346, Spring 2013
1) We have
4/7, x = 2,
pX (x) = 3/7, x = 1,
0,
otherwise.
It follows that
4/9, y = 3,
pX,Y (1, y )
(1 + y )/21
1+y
3/9, y = 2,
pY |X (y |1) =
=
Icfw_1,2,3 (y ) =
Icfw_1,2,3 (y ) =
2/9, y = 1,
pX (1)
3/7
9
0,
otherwise.
Using
Homework #4 STAT346 Section 003
Question 1:
(a) Prove that
n
n
n
n n
k n
= 0.
+ + (1)
+
+ (1)
n
k
0
1
2
(b) Evaluate the following sum:
n
1 n
1 n
1
n
+
+
+ +
.
0
2 1
3 2
n+1 n
n+1
1 n
1
Hint: i+1
i = n+1 i+1 .
Question 2: A big urn contains
Homework #2 STAT346 Section 003
Question 1: Suppose that in a metropolitan area 65% of the car accidents occur during the
day and 75% of the car accidents occur in the city. If only 20% of the car accidents occur
outside the city during the day, then
(a)
Homework #5 STAT346 Section 003
Question 1: Urn I contains 25 white and 20 black balls. Urn II contains 15 white and 10
black balls. An urn is selected at random and one of its balls is drawn randomly and observed
to be black and then returned to the same
Homework #3 STAT346 Section 003
Question 1: A campus telephone extension has four digits.
(a) How many different extensions exist? Of these, how many include at least a 0?
(b) How many different extensions with no repeated digits exist? Of these with no r