Let ?1 and ?2 be two independent random variables with uniform distribution on [0, 1].
1. Write down the joint cumulative distribution function and joint probability
density function of ?1 + ?2 and ?1?2.
2. Write down the covariance between ?1 + ?2 and ?1?2.
3. Let ? be the largest magnitude (absolute value) of a root of the equation
?^2 − ?1? + ?2 = 0. Let ? be the random event that says that the
equation ?^2 −?1? +?2 = 0 has no real roots. Find the conditional c.d.f.
of ? when ? is false,

Let ?1 and ?2 be two independent random variables with uniform distribution on [0, 1].

1. Write down the joint cumulative distribution function and joint probability density function of ?1 + ?2 and ?1?2.

2. Write down the covariance between ?1 + ?2 and ?1?2.

3. Let ? be the largest magnitude (absolute value) of a root of the equation ? 2 − ?1? + ?2 = 0. Let ? be the random event which says that the equation ? 2 − ?1? + ?2 = 0 has no real roots. Find the conditional c.d.f. of ? when ? is true

Valuation Methods Exercise

Assume all years' cash flows will grow 4%

Excel 1

Suppose 2 boxes:
- Box #1 contains:
+ 5 Black balls.
+ 6 White balls.
- Box #2 contains:
+ 8 Black balls.
+ 4 White balls.
- Two balls are drawn from Box #2 and placed into Box #1.
- A ball is drawn from Box #1.

1- What is the probability that the ball drawn from Box #1 is White?
2- Suppose that the ball drawn from Box #1 is White, then what is the probability that at least one White
ball was transferred to Box #1?

Suppose 2 boxes:
- Box #1 contains:
+ 5 Black balls.
+ 6 White balls.
- Box #2 contains:
+ 8 Black balls.
+ 4 White balls.
- Two balls are drawn from Box #2 and placed into Box #1.
- A ball is drawn from Box #1.

1- What is the probability that the ball drawn from Box #1 is White?
2- Suppose that the ball drawn from Box #1 is White, then what is the probability that at least one White
ball was transferred to Box #1?

Answer true or false for each statement and give a logically valid explanation to support your answer (you can give a counterexample if it is false, or make a proof if it is true).

a. The multiplication of two irrational numbers is an irrational number.

b. The set {( 1−n/ n , 8n+1/ n ) |n ∈ N } is an open covering of (−1, 8].

c. The sequence {( 1−3n /n + (−1)ⁿ ) |n ∈ N } has a sub-sequence that converges to -4.

d. The sum of the Lebesgue measure of the sets A = C ∪ [−5, 1), and B = Q ∩ [4, 10] is 6, where C is the Cantor ternary set, and Q is the rational numbers set.

e. Find the inf and sup of the set A = {x ∈ R : x ² − 4x ≤ 20 − 5x} are -5 and 4 respectively.

f. The intersection set for the following collection of intervals is open. {(−1/n − 2, 3 + 1/n), n ∈ N} .

An irrigation experiment was conducted in a randomized complete block design in a Lavencia orange grove. Six irrigation treatments were used in eight blocks of trees. The data below are the weight in pounds of harvested fruit from each plot.

a. Consider the flood irrigation method to be the standard practice. Use the Dunnett method to test the difference between the flood irrigation and each of the other methods

b. Calculate the F-statistic assuming a completely randomized one-way design. How different are the results?

c. Check the assumptions from part b. Make sure to include any plots in your solutions

Marvin’s Milk Farm produces milk and sells it in a perfectly competitive market at $3 per bottle. The following table shows Marvin's weekly total and marginal product schedules, using labor and capital. Assume that labor and capital may be used independently; that is, one is not needed for the other factor to be productive. Therefore, the total amount of milk that Marvin's produces is obtained by adding together the amount of milk produced by labor and the amount of milk produced by capital. The table also shows total revenue and marginal revenue products (MRP) of labor and capital. Finally, assume that Marvin's Milk Farm is a factor price taker in the labor and capital in markets. Labor costs $36 per week, and capital costs $72 per week.

If Marvin’s Milk Farm wants to produce 70 bottles of milk per week, the least-cost combination of labor and capital is (1, 2, 3, or 4 units) of labor and (1, 2, 3, or 4 units) of capital.

The profit-maximizing combination of resources is (1, 2, 3, or 4 units) of labor and (1, 2, 3, or 4 units) of capital.

The profit-maximizing combination contains (the same amounts of labor and capital as, fewer units of labor and capital than, or more units of labor and capital than) the least-cost combination to produce 70 bottles of milk.

No essays or confusing explanations, please. Just answers.

Exact Photo Service purchased a new color printer at the beginning of Year 1 for $39,700. The printer is expected to have a four-year useful life and a $3,700 salvage value. The expected print production is estimated at $1,770,500 pages. Actual print production for the four years was as follows:

The printer was sold at the end of Year 4 for $4,100.

b. Compute the depreciation expense for each of the four years, using units-of-production depreciation.

Depreciation Expense

Year 1

Year 2

Year 3

Year 4

Total accumulated depreciation$0

Exact Photo Service purchased a new color printer at the beginning of Year 1 for $39,700. The printer is expected to have a four-year useful life and a $3,700 salvage value. The expected print production is estimated at $1,770,500 pages. Actual print production for the four years was as follows:

The printer was sold at the end of Year 4 for $4,100.

c. Calculate the amount of gain or loss from the sale of the asset under each of the depreciation methods.

DDB =

Units-of-production=

solve completely(complementary function and particular integral) pdes

1 (1+D^2-D')z = -8siny

2 (D-D'-D'^2)z =132xy^10+504x^2y^6

3 (D^2-4DD' +4D'^2)z = x + y

4 (D^2-2DD'+ D'^2)z = e^2x-2y+x^2