The joint probability density function for two continuous random variables ?? and ?? is
??(??, ??) = ??(3??2 − ??), 0 < ?? < 1, 0 < ?? < 1.
Answer the following:
(a) Find the value of ?? such that ??(??, ??) is a valid probability density function.
(b) Find the marginal probability density function for ??, ??(??).
(c) Find the marginal probability density function for ??, ??(??).
(d) Find the conditional probability density function for ??|?? = ??, ??(??|??).
(e) Based on your answers from (b) to (d) only, are ?? and ?? independent?
(f) Calculate the covariance between ?? and �

The probability that Michael misses a free throw shot is .1. If he goes to the line to shoot three free throws (due to a foul on a three-point shot),

a) What is the probability that Michael misses all three shots? What assumptions did you make in order to calculate this probability?

b) What is the probability that Michael makes at least one of the three shots?

c) What is the probability that Michael makes the first shot but not the second or third?

1. Chris and Pat both work independently on the same computer program. The probability Chris’ program works is 10% and the probability Pat’s program works is 15%. What is the probability exactly one of their programs works?  

2. Terry and pat both play independently with the same computer game. The probability Terry will score a victory is 30% and the probaabiltythat Pat wwill score a victory is 25%. What is the probability that at least one of them scores a victory?

The fill amount of bottles of a soft drink is normally​ distributed, with a mean of 1.01.0 literliter and a standard deviation of 0.040.04 liter. Suppose you select a random sample of 2525 bottles. a. What is the probability that the sample mean will be between 0.990.99 and 1.01.0 literliter​? b. What is the probability that the sample mean will be below 0.980.98 literliter​? c. What is the probability that the sample mean will be greater than 1.011.01 ​liters? d. The probability is 9999​% that the sample mean amount of soft drink will be at least how​ much? e. The probability is 9999​% that the sample mean amount of soft drink will be between which two values​ (symmetrically distributed around the​ mean)? a. The probability is nothing. ​(Round to three decimal places as​ needed.) b. The probability is nothing. ​(Round to three decimal places as​ needed.) c. The probability is nothing. ​(Round to three decimal places as​ needed.) d. There is a 9999​% probability that the sample mean amount of soft drink will be at least nothing ​liter(s). ​(Round to three decimal places as​ needed.) e. There is a 9999​% probability that the sample mean amount of soft drink will be between nothing ​liter(s) and nothing ​liter(s). ​(Round to three decimal places as needed. Use ascending​ order.)

PLEASE SHOW ME HOW TO DO IT IN EXCEL, THANKS

1. A man claims to have extrasensory perception (ESP). As a test, a fair coin is flipped 28 times, and the man is asked to predict the outcome in advance. He gets 22 out of 28 correct. What is the probability that he would have done at least this well if he had no ESP?
Probability =

2. A careless university student leaves her iClicker device behind with probability 1/4 each time she attends a class. She sets out with her iClicker device to attend 5 different classes (each class is in a different lecture theatre).

a) If she arrives home without her iClicker device (after attending 5 classes), what is the probability (to 3 significant figures) that she left it in the 5th class? Hint: This is a conditional probability! Probability =

b) If she arrives home without her iClicker device and she is sure she has the iClicker device after leaving the first class, what is the probability (to 3 SIGNIFICANT figures) that she left it in the 5th class? Hint: This is a conditional probability! Probability =

Problem 12-18 (Algorithmic)

A building contractor is preparing a bid on a new construction project. Two other contractors will be submitting bids for the same project. Based on past bidding practices, bids from the other contractors can be described by the following probability distributions:

If required, round your answers to three decimal places.

1. If the building contractor submits a bid of $670,000, what is the probability that the building contractor will obtain the bid? Use an Excel worksheet to simulate 1,000 trials of the contract bidding process.
   
   The probability of winning the bid of $670,000 =

b. The building contractor is also considering bids of 695,000 and $705,000. If the building contractor would like to bid such that the probability of winning the bid is about 0.7, what bid would you recommend? Repeat the simulation process with bids of $695,000 and $705,000 to justify your recommendation.

The probability of winning the bid of $695,000 =

The probability of winning the bid of $705,000 =

The recommendation would be to choose the bid of _______

Consider the following pseudo-code:

/* Global memory area accessible by threads */
#define N 100
struct frame *emptyStack[N];
struct frame *displayQueue[N];

int main() {
  /*

  ** Initialise by allocating the memory for N frames
  ** And place the N frame addresses into the
  ** empty Stack array
  */

  Initialise();
  thread_t tid1, tid2;

  threadCreate(&tid1, GetFrame);
  threadCreate(&tid2, ShowFrame);

  sleep(300);
}

GetFrame() {
  struct frame *frame;
  struct frame local;

  while (1) {
    CameraGrab(&local);  /* get a frame from the camera store it in local */

    frame = Pop(); /* pop an empty-frame address from the empty stack */

    CopyFrame(&local, frame); /* copy data from the local frame to the frame address */

    Enqueue(frame); /* push the frame address to the display queue */
}

}

ShowFrame() {
  struct frame *frame;
  struct frame local;
  struct frame filtered;

  while (1) {
    frame=Dequeue(); /* pop the leading full frame from the full queue */

    CopyFrame(frame, &local); /* copy data to the local frame */

    Push(frame);          /* push the frame address to the empty stack */

    Solarise(&filtered, &local); /* Modify the image */

    VideoDisplay(&filtered);     /* display the image */
  }

}

This program creates two threads, one calls GetFrame(), which continually grabs frames from a camera, and the other thread callsShowFrame(), which continually displays the frames (after modification). When the program starts the emptyStack contains the addresses of N empty frames in memory.

The process runs for 5 minutes displaying the contents from the camera.

The procedures Pop() and Push() are maintaining the list of frame addresses on the empty stack. Pop() removes a frame memory address from the empty stack, and Push() adds a memory address to the empty stack.

The procedures Dequeue() and Enqueue() are maintaining the list of frame memory addresses in the display queue in display order. Dequeue() removes the memory address of the next frame to display from the display queue, and Enqueue() adds a memory address to the end of the display queue.

The stack and the queue are the same size, and are large enough to contain all available frame memory addresses.

1. Without including synchronisation code problems will occur. Discuss what could potentially go wrong?

2. Identify the critical sections in the above pseudo-code.

3. Modify the above pseudo-code using semaphores only, to ensure that problems will not occur.

   Hint: this is a variation on the Producer-Consumer problem and will require similar semaphores.

  Boardman Gases and Chemicals is a supplier of highly purified gases to semiconductor manufacturers. A large chip producer has asked Boardman to build a new gas production facility close to an existing semiconductor plant. Once the new gas plant is in​ place, Boardman will be the exclusive supplier for that semiconductor fabrication plant for the subsequent 10 years. Boardman is considering one of two plant designs. The first is ​ Boardman's "standard" plant which will cost 38.3 million to build. The second is for a​ "custom" plant which will cost $53.3 million to build. The custom plant will allow Boardman to produce the highly specialized gases required for an emergency semiconductor manufacturing process. Boardman estimates that its client will order ​$11.7 million of product per year if the standard plant is​ constructed, but if the custom design is put in​ place, Boardman expects to sell ​$17.1 million worth of product annually to its client. Boardman has enough money to build either type of​ plant, and, in the absence of risk​ differences, accepts the project with the highest NPV. The cost of capital is 16.6​%.

A. Find the NPV for each project. Are the projects​ acceptable?

B. Find the breakeven cash inflow for each project.

C. The firm has estimated the probabilities of achieving various ranges of cash inflows for the two projects, as shown in the table..... What is the probability that each project will achieve the breakeven cash inflow found in part B?

Probability of achieving cash inflows in given range

Range of cash inflows Standard plant Custom Plant

$0 to 5 0% 5%

5 to 8 10 10

8 to 11 60 15

11 to 14 25   25

14 to 17 5    20

17 to 20 0 15

above 20 0 10

d. Which project is more​ risky? Which project has the potentially higher​ NPV? Discuss the​ risk-return trade-offs of the two projects.

e. If the firm wished to minimize losses​ (that is, NPV less than $ 0NPV<$0​), which project would you​ recommend? Which would you recommend if the goal was achieving a higher​ NPV?

Boardman Gases and Chemicals is a supplier of highly purified gases to semiconductor manufacturers. A large chip producer has asked Boardman to build a new gas production facility close to an existing semiconductor plant. Once the new gas plant is in​ place, Boardman will be the exclusive supplier for that semiconductor fabrication plant for the subsequent 10 years. Boardman is considering one of two plant designs. The first is ​ Boardman's "standard" plant which will cost ​$39.1 million to build. The second is for a​ "custom" plant which will cost ​$54.1 million to build. The custom plant will allow Boardman to produce the highly specialized gases required for an emergency semiconductor manufacturing process. Boardman estimates that its client will order​$12.3 million of product per year if the standard plant is​ constructed, but if the custom design is put in​ place, Boardman expects to sell ​$16.7 million worth of product annually to its client. Boardman has enough money to build either type of​ plant, and, in the absence of risk​ differences, accepts the project with the highest NPV. The cost of capital is 17.2%.

a. Find the NPV for each project. Are the projects​ acceptable?

b. Find the breakeven cash inflow for each project.

c. The firm has estimated the probabilities of achieving various ranges of cash inflows for the two​ projects, as shown in the table. What is the probability that each project will achieve the breakeven cash inflow found in part ​(b​)​?

d. Which project is more​ risky? Which project has the potentially higher​ NPV? Discuss the​ risk-return trade-offs of the two projects.

e. If the firm wished to minimize losses​ (that is, NPV < $0​), which project would you​ recommend? Which would you recommend if the goal was achieving a higher​ NPV?