1. Rolling two D20

Consider what hapens when we roll two 20-sided dice d1 and d2 (so the sample space is S={(d1,d2):d1,d2∈{1,2,3,…,20}} and Pr(ω)=1/|S| for each ω∈S). Consider the following events:

• A is the event "d1=13"
• B is the event "d1+d2=15"
• C is the event "d1+d2=21"

Use the definitions of independence and conditional probability to answer these two questions:

1. Are the events A and B independent?
2. Are the events A and C independent?

A stock sells for $84 and pays a continuously compounded 3% dividend. The continuously compounded risk-free rate is 5%.

a. What is the price of a pre-paid forward contract for one share to be delivered six months (.5 year) from today?

b. What is the price of a forward contract that expires six months from today?

c.Describe the transactions you would undertake to use the stock and bonds (borrowing and lending) to construct a synthetic long forward contract for one share of stock.

Question: Linda has 5 weeks to prepare for her CSCA67 final. Her friend has volunteered to help her for either 15min or 30min every day until the test but not for more than 15 hours total. Show that during some period of consecutive days, Linda and her help will study for exactly 8 3/4 hours.

Answer: We can solve this by letting ai represent the number of quarter hours Linda studies on day i. Then there are 5x7 = 35 days that Linda studies. Now we define 35 sums: s1 = a1, s2 = a1 +a2, s3 = a1 +a2 +a3. Then if one of these sums equals 8 ∗ 4+3 = 35 quarter hours, we are done. If not, then there are 35 sums (pigeons) and we set our holes to be the possible remainders for each sum when divided by 35, we have the values from 1..34 or 34 holes. Therefore there are two pigeons in one hole, ie, two sums that when divided by 35 have the same remainder. If we subtract the smaller sum from the larger we get a continuous subset of days (by the way we designed the si and this difference must be divisible by 35. Since no sum is larger than 60 and the difference is a multiple of 35, this multiple cannot be larger than 1. Therefore we have a set of consecutive days totally 8 3 4 hours.

My question: Can someone help me to under stand and solve this question in proper way by using php. I dont understand how answer says, "35 sums (pigeons) and we set our holes to be the possible remainders for each sum when divided by 35, we have the values from 1..34 or 34 holes"

Prove that L = {a + b √ 5i | a, b ∈ Q} is a field containing the roots of x² + 5. Moreover, prove that if Q ⊆ K ⊆ C is a field containing the roots of x² + 5, then L ⊆ K.

A country uses bronze coins with denominations of 1 peso, 2 pesos, 3

pesos, and 5 pesos, silver coins of denominations of 2 pesos and 5

pesos, and uses gold coins with denominations of 5 pesos, 10 pesos,

and 20 pesos. Show work.

a ) Find a recurrence relation for the number of ways to pay a bill of n

pesos if the order in which the coins are paid matters

b) How many ways are there to pay a bill of 6 pesos

if the order in which the coins are

paid matters?

Use matlab to plot Taylor approximation for f(x) for x near 0 Given f(x) = exp(x)

Use subpots to plot and its absolute and realative erros for N= 1,2,3

PLease give matlab code for Taylor and explain in detail, Thank you

If you had a routine for computing all of the eigenvalues of a nonsymmetric matrix, how could you use it to compute the roots of any polynomial?

A population subject to the Allee effect will decrease when population numbers are either very low (due to inability to find a mate) or very high (due to overcrowding), but will increase when population numbers are at intermediate levels (due to successful reproduction and not too much overcrowding). A model for a population subject to an Allee effect is given by

x_t+1= (4x²_t)/(3+x²_t), x_t is greater than and equal to 0

where ?_t is the population number in year t (measured in hundreds of individuals.

(a) Determine the equilibria.

(b) Use the Stability Criterion to classify the equilibria as asymptotically stable or unstable.

(c) Use cobwebbing to illustrate the dynamics of the difference equation for ?₀=0.5, ?₀=0.5, ?₀=1.5 and ?₀=4(it is OK to plot all three on the same diagram).

(d) Explain the significance of all equilibria and their stability in terms of the population size in the long run.

A company manufactures and sells x television sets per month. The monthly cost and​ price-demand equations are ​C(x)equals72 comma 000 plus 70 x and p left parenthesis x right parenthesis equals 300 minus StartFraction x Over 20 EndFraction ​, 0less than or equalsxless than or equals6000. ​(A) Find the maximum revenue. ​(B) Find the maximum​ profit, the production level that will realize the maximum​ profit, and the price the company should charge for each television set. ​(C) If the government decides to tax the company ​$4 for each set it​ produces, how many sets should the company manufacture each month to maximize its​ profit? What is the maximum​ profit? What should the company charge for each​ set? ​(A) The maximum revenue is ​$ nothing. ​(Type an integer or a​ decimal.) ​(B) The maximum profit is ​$ nothing when nothing sets are manufactured and sold for ​$ nothing each. ​(Type integers or​ decimals.) ​(C) When each set is taxed at ​$4​, the maximum profit is ​$ nothing when nothing sets are manufactured and sold for ​$ nothing each. ​(Type integers or​ decimals.)

Let X = {1, 2, 3, 4, 5, 6} and let ∼ be given by {(1, 1),(2, 2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 3),(1, 5),(2, 4),(3, 1),(3, 5), (4, 2),(5, 1),(5, 3)}.

Is ∼ an equivalence relation? If yes, write down X/ ∼ .

what is the value of ∫∞−∞ δ(t+2)e^(−2t)u(t)dt?

A tank contains 2140 L of pure water. A solution that contains 0.01 kg of sugar per liter enters a tank at the rate 6 L/min. The solution is mixed and drains from the tank at the same rate.

(a) How much sugar is in the tank initially?

(b) Find the amount of sugar in the tank after t minutes.

(c) Find the concentration of sugar (kg/L) in the solution in the tank after 51 minutes.

4. Let A be the 6*6 diagonal matrix below. For each eigenvalue, compute the multiplicity of λ as a root of the characteristic polynomial and compare it to the dimension of the eigenspace Eλ.

(x 0 0 0 0 0 0 x 0 0 0 0 0 0 y 0 0 0 0 0 0 x 0 0 0 0 0 0 z 0 0 0 0 0 0 x)

5. Let A be an 3*3 upper triangular matrix with all diagonal elements equal, such as (3 4 -2 0 3 12 0 0 3) Prove that A is diagonalizable if and only if A is a scalar times the identity matrix.

(a) Devise and then describe a method to systematically list all rational numbers between 0 and 1.

(b) Using your answer to part (a), show that the set of rationals between 0 and 1 has the same cardinality as the set of natural numbers.