Business Statistics 41000: : Homework # 3
Drew Creal
February 3, 2014
Remarks: These questions cover Lectures #5 are due at the beginning of Lecture # 7.
Question # 1. Understanding how to read a pdf
Below is a plot of the pdf of some continuous random variable.
0.150
0.125
0.100
0.075
0.050
0.025
0
Note:
(a)
f (x) = 0
2
for all
4
x
6
8
10
values less than 0.
What is the probability of being less than 0 ?
1
12
14
16
18
20
(b)
What is the probability of being greater than 25?
(c)
The probability of being less than 5.3 is:
(i)
(d)
0.1
(ii)
0.95
(iii)
0.3
(iv)
0
The probability of being between 5.3 and 10.6 is:
(i)
0.1
(ii)
0.98
(iii)
0.6
(iv)
0.2
Question # 2. The Uniform Distribution
Go back to the cloth cutting example in the notes. Let
X
equal the length of the remnant of
cloth in inches. Suppose our remnant is equally likely to have any length between 0 and 3 inches
but cannot be bigger than 3 inches.
(a)
Sketch the pdf. In particular, what is the height of the pdf ?
(b)
What is the probability that the remnant is less than one inch in length?
(c)
What is the probability that the remnant is greater than 1.5 inches in length?
Question # 3. The Standard Normal Distribution
Suppose
Z
has the standard normal distribution, i.e.
Z ∼ N (0, 1).
question using only simple arithmetic. You do not need a computer.
(a)
What is
P (Z > 1.96)?
(b)
What is
P (Z > −1.96)?
(c)
What is
P (0 < Z < 1.96)?
(d)
What is
P (−1 < Z < 1.96)?
Hint: You can answer this
Question # 4. The Normal Distribution
Below are the p.d.f.s for four random variables
X1
0.75
0.50
0.25
0.25
−2
0
2
deviation
−4
0.50
0.25
0.25
Xi
σ
−2
0
2
has a normal distribution with mean
(a)
What is the mean and variance of
X1 ?
(b)
What is the mean and variance of
X2 ?
(c)
What is the mean and variance of
X3 ?
(d)
What is the mean and variance of
X4 ?
0
2
4
X4
4
either 1 or 0.5.
−2
0.75
0.50
−4
i = 1, 2, 3, 4.
X2
4
X3
0.75
for
0.75
0.50
−4
Each
Xi
−4
µ
−2
0
2
4
equal to one of -1, 1, 2, or 0, and standard
Question # 5. The Normal Distribution
Suppose
R is a random variable representing returns on an asset next month and R ∼ N (0.05, 0.01).
(a)
Give an interval such that
(b)
What is
P (R > 0.05)?
(c)
What is
P (R > 0.25)?
(d)
What is
P (−0.05 < R < 0.15)?
(e)
What is
P (R < 0.05)?
R
has a 95% chance of being in that interval?
Question # 6. The Uniform Distribution
Suppose
X ∼ U (a, b),
i.e. it is uniform on the interval
(a)
Find a formula for the c.d.f. of
(b)
If
X ∼ U (0, 1),
X.
what is the c.d.f.
F (x)?
(a, b).