(a) Find the expected
value of the random variable
X.
(b) Find the variance of the random variable
X.
(c) Describe the distribution of the random variable Y .
(d)


Exercise 4. Suppose that the daily rate of return on a certain stock is normally
distributed with a mean of µ = 0.23% and a standard deviation
of σ = 2.54%. What is the probability that the stock’s price will increase
by more than 5.31% on any given day next week?
Exercise 5. Suppose that the daily rate of return on a certain stock is normally
distributed with a mean of µ = 0.23% and a standard
deviation of σ = 2.54%.
(a)
What is the probability that the stock’s
price will increase
by more than 2% on any given
day next week?
(b) What is the probability that the stock’s
price will decrease
by more than 1% on any given
day next week?
Exercise 6. Let X be a random
variable and let c ∈ R be a real number. Demonstrate that the expectation operator E satisfies E [cX] = c · E [X].
Exercise 7. Let X be a random variable and let c ∈ R be a real number. Demonstrate that the variance
operator V satisfies V [cX] = c^{2} · V [X].
Exercise 8. The number of defaults in one year within a certain portfolio
of bonds is found to be a Poisson random
variable with parameter
λ
= 7 (i.e., the portfolio has an expectation of 7 defaults per year).
(a)
Find the probability that the bond portfolio will have fewer than 3 defaults during the upcoming
year.
(b) Describe the distribution of the continuous random variable representing the time between
successive defaults.
(c)
Find the expected time between successive
defaults.
(d) Calculate the probability of there being less than 6 months between two successive defaults.
(a) Find the probability that X is less than or equal to 0.1.
(b)

and identically distributed as X so that Y
return.
=∆ X + X + X + X represents the stock’s fourday logarithmic
(i) Describe the distribution of the continuous random variable Y .
(ii) Calculate the probability that Y is less than or equal to 0.1.
1
(c)

0
representing the ratio
of tomorrow’s price
to
today’s
price.
(d)

What is the expected value of the random variable^{ }^{S}1 ?
0
(e)

What is the variance of the random variable^{ S}1 ?
0
Exercise 10. Let the dynamics X_{i}, i = 1, 2, . . . , d be
independently and identically distributed as Z ∼
N (0, 1). One approach for modeling the shortterm
interest
rate
r_{t} at any time t is given by defining
r =∆ X^{2} + X^{2} + . . . + X^{2}.
t 1 2 d
(a)
Describe the distribution of the continuous random variable r_{t}.
(b) Find the probability that r_{t} ∈ (0, 0.02] if d = 3.
(c) Find the probability that r_{t} ∈ (0, 0.02] if d = 7.
Exercise 11 (BONUS). Prove
that the variance
of the sum of two independent random
variables is equal
to the sum of their individual variances.
This Question
Hasn’t Been Answered Yet! Do You Want an Accurate, Detailed, and Original Model
Answer for This Question?
Copyright © 2012  2023 Apaxresearchers  All Rights Reserved.