(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 .
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] = c2 · 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.
and identically distributed as X so that Y
=∆ X + X + X + X represents the stock’s four-day 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.
representing the ratio of tomorrow’s price to
What is the expected value of the random variable S1 ?
What is the variance of the random variable S1 ?
Exercise 10. Let the dynamics Xi, i = 1, 2, . . . , d be independently and identically distributed as Z ∼
N (0, 1). One approach for modeling the short-term interest rate rt at any time t is given by defining
r =∆ X2 + X2 + . . . + X2.
t 1 2 d
(a) Describe the distribution of the continuous random variable rt.
(b) Find the probability that rt ∈ (0, 0.02] if d = 3.
(c) Find the probability that rt ∈ (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.
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