Check in each case whether the given function can serve as the probability distribution of an appropriate random variable. (a) f(x) = (2 x)/4 for x = 0, 1, 2; (b) f(x) = x - 2/9 for x = 1, 2, 3, 4, 5, 6; (c) f(x) = x^2 - 6x + 9/10 for x = 1, 2, 3, 4, 5; (d) f(x) = x^2 - 6x + 8/5 for x = 1, 2, 3, 4, 5.

The following table represents a plan for a project:

b. Indicate the critical path.

c. What is the expected completion time for the project? (Round your answer to 2 decimal places.)

Expected completion time            days

d. You can accomplish any one of the following at an additional cost of $2,500 and if you will save $1,700 for each day that the earliest completion time is reduced, which action, if any, would you choose?

1. Reduce job 2 by three days.

2. Reduce job 6 by two days.

3. Reduce job 10 by two days.

e. What is the probability that the project will take more than 29 days to complete? (Round your answer to 2 decimal places.)

Probability           

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3} and transition probability matrix (pij ). Let τi = min{n ≥ 1 : Xn = i}, i = 0, 1, 2, 3. Define Bij = {Xτj = i}. Is Bij ∈ σ(X0, · · · , Xτj ) ? Q2. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, X0 = 0, and transition probability matrix (pij ) given by   2 3 1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2   Let Bij as in Q1. Compute P(Xτ0+2 = 2|B00).

Q4. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2} and transition probability matrix (pij ) given by   2 3 1 3 0 1 3 2 3 0 0 1 4 3 4   Let τ2 = inf{n ≥ 1 : Xn = 2} Find P{τ2 < ∞|X0 = 2}. Q5. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, and transition probability matrix (pij ) given by   2 3 1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2   Determine all recurrent states. 1

1) Rank the following titrations in order of increasing pH at the halfway point to equivalence (1 = lowest pH and 5 = highest pH).

1 2 3 4 5  200.0 mL of 0.100 M H₂NNH₂ (K_b = 3.0 x 10^-6) by 0.100 M HCl

1 2 3 4 5  100.0 mL of 0.100 M HC₃H₅O₂ (K_a = 1.3 x 10^-5) by 0.100 M NaOH

1 2 3 4 5  100.0 mL of 0.100 M HF (K_a = 7.2 x 10^-4) by 0.100 M NaOH

1 2 3 4 5  200.0 mL of 0.100 M (C₂H₅)₂NH (K_b = 1.3 x 10^-3) by 0.100 M HCl

1 2 3 4 5  100.0 mL of 0.100 M HI by 0.100 M NaOH

2) Rank the following titrations in order of increasing pH at the equivalence point of the titration (1 = lowest pH and 5 = highest pH).

1 2 3 4 5  100.0 mL of 0.100 M HC₃H₅O₂ (K_a = 1.3 x 10^-5) by 0.100 M NaOH

1 2 3 4 5  200.0 mL of 0.100 M (C₂H₅)₂NH (K_b = 1.3 x 10^-3) by 0.100 M HCl

1 2 3 4 5  200.0 mL of 0.100 M H₂NNH₂ (K_b = 3.0 x 10^-6) by 0.100 M HCl

1 2 3 4 5  100.0 mL of 0.100 M HF (K_a = 7.2 x 10^-4) by 0.100 M NaOH

1 2 3 4 5  100.0 mL of 0.100 M KOH by 0.100 M HCl

Write a class named Rat (to simulate rational or fraction number) such that it works as in the main below.

int main()

{

Rat a(3, 4), b(1, 2);

a.print(); // output: 3/4

b.print(); // output: 1/2

Rat c = a.add(b);

c.print(); // output: 5/4

// Hint: 3/4 + 1/2 = 6/8 + 4/8 = 10/8 = 5/4

Rat d = a.multiply(b);

d.print(); // output: 3/8

// Hint: 3/4 x 1/2 = 3/8

return 0;

}

provide a C code (only C please) that gives the output below:

************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials. *
* 4. Displaying polynomials *
* 5. Clearing polynomials. *
* 6. Quit. *

***********************************

Select the option (1 through 6): 7

You should not be in this class!

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials.   *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 4

Left Poly Pointer: 0

Right Poly Pointer: 0

Resulting Poly Pointer: 0

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 1

/* Performing the required task(s) and your code must ALSO print

1. Description/explanation of the method or approach that you

use to create 2 polynomials; and
2. The listing of all functions involved in the process.

*/

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 4

Left Poly Pointer: SOME NONE ZERO ADDRESS and DISPLAYING Poly 1/1x2 + 3/4x + 5/12

Right Poly Pointer: SOME NONE ZERO ADDRESS and DISPLAYING Poly 1/1x4 – 3/7x2 + 4/9x + 2/11

Resulting Poly Pointer: 0

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 2

/* Performing the required task(s) and your code must ALSO print

1. Description/explanation of the method or approach that you

use to add 2 polynomials; and
2. The listing of all functions involved in the process.

*/

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

***********************************

Select the option (1 through 6): 4

Left Poly Pointer: SOME NONE ZERO ADDRESS and DISPLAYING Poly 1/1x2 + 3/4x + 5/12

Right Poly Pointer: SOME NONE ZERO ADDRESS and DISPLAYING Poly 1/1x4 – 3/7x2 + 4/9x + 2/11

Resulting Poly Pointer: SOME NONE ZERO ADDRESS and DISPLAYING Poly 1/1x4 – 3/7x2 + 43/36x + 79/132

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 3

/* Performing the required task(s) and your code must ALSO print

1. Description/explanation of the method or approach that you

use to multiply 2 polynomials; and
2. The listing of all functions involved in the process.

*/

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 4

Left Poly Pointer: SOME NONE ZERO ADDRESS and DISPLAYING Poly 1/1x2 + 3/4x + 5/12

Right Poly Pointer: SOME NONE ZERO ADDRESS and DISPLAYING Poly 1/1x4 – 3/7x2 + 4/9x + 2/11

Resulting Poly Pointer: SOME NONE ZERO ADDRESS and DISPLAYING Poly 1/1x6 + 3/4x5 – 1/84x4 + 31/252x3 + 871/924x2 + 191/594x + 5/66

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 5

/* Releasing selected polynomial(s)
For example, clearing and releasing left polynomial

*/

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 4

Left Poly Pointer: 0
Right Poly Pointer: SOME NONE ZERO ADDRESS and DISPLAYING Poly

1/1x4 – 3/7x2 + 4/9x + 2/11

Resulting Poly Pointer: 0

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 5

/* Releasing selected polynomial(s)
For example, clearing and releasing right polynomial

*/

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 4

Left Poly Pointer: 0

Right Poly Pointer: 0

Resulting Poly Pointer: 0

*************************************
*         Menu HW #4 *
* POLYNOMIAL OPERATIONS *
* 1. Creating polynomials *
* 2. Adding polynomials *
* 3. Multiplying polynomials *
* 4. Displaying polynomials. *
* 5. Clearing polynomials *
* 6. Quit *

***********************************

Select the option (1 through 6): 6

Polynomials -- Having Fun!

Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom. The four quantum numbers, the principal quantum number (n),(n), the angular momentum quantum number (?),(?), the magnetic quantum number (m?),(m?), and the spin quantum number (ms)(ms) have strict rules which govern the possible values.

Identify all allowable combinations of quantum numbers for an electron.

1. n=3, ?=-2, m? = -2, ms = +1/2

2. n=3, ?=2, m? = -2, ms = -1/2

3. n=5, ?=4, m? = -1, ms = -1/2

4. n=2, ?=0, m? = 0, ms =1

5. n=1, ?=1, m? = -1, ma = -1/2

6. n=4, ?=3 m? = 4, ms = -1/2

Deadlock Avoidance using Banker’s Algorithm

Q2: Use the following information and complete the Table, also write down the safe sequence if exist?

Three Resources (R1=4, R2=9, R3=11)

1.) Make a multiple regression model using these potential numerical predictor variables and, at most, one categorical dummy variable.

2.)Write the sample multiple regression equation for the “final best” model you have developed.

3.) Look at the set of residual plots, cut and paste them into the report, and briefly comment on the appropriateness of your fitted model.