1. A company has a cash portfolio measured in millions. The drift is 0.1 per month, variance is 0.16per month. The initial cash is 2.0. a) Find the probability distribution after 6 months and after 1 year. b) Find the probability of a negative cash position at the end of 6 months and the end of 1 year. c) At what time in the future is the probability of a negative distribution greatest.

Answer:

Step-by-step explanation:

From the information given:

The probability distribution at the end of 6 months is determined as follows:

After 6 months;

Mean of probability distribution =  value of Initial cash  + [tex]\alpha[/tex]T  

=2.0 +(0.1 × 6)

=2.6

After 6 months;

The probability distribution's standard deviation is estimated by using the following formula:

Standard deviation:

[tex]= b\sqrt{T}[/tex]

[tex]= 0.4 \times \sqrt{6}[/tex]

= 0.9798

Hence, after 6 months;

The company's cash position is supposed to be allocated monthly, with the following expenses.

Mean                            2.6

Standard deviation     0.9798

Variance                      0.96

After 12 months, the probability distribution is as follows:

Mean = value of Initial cash  + [tex]\alpha[/tex]T

= 2.0 +(0.1 × 12)

= 3.2

The standard deviation is:

The standard deviation of probability distribution = [tex]b \sqrt{T}[/tex]

[tex]= 0.4 \times \sqrt{12}[/tex]

= 1.3856

Hence, after 6 months;

The company's cash position is supposed to be allocated monthly, with the following expenses.

Mean                            3.2

Sandard deviation      1.3856

Variance                      1.92

b)  

in 6-month distribution, the probability of the negative value of the cash position is as follows.

Now, for us to find the negative cash distribution;

We need to estimate the z -scores value.

The z-score inform us greatly on the concept of how far a particular data point is from the mean.

For a normal distribution;

[tex]z = \dfrac{x-\mu}{\sigma}[/tex]

Here;

the value of x = zero as a result that if it exceeds zero. the cash position will be negative.

∴

[tex]z = \dfrac{x-\mu}{\sigma}[/tex]

[tex]z = \dfrac{0 - 2.6}{0.9798}[/tex]

[tex]z = -2.6536[/tex]

Using the standard distribution tables, it is now possible to calculate that the likelihood N(-2.65) equals 0.004 or 0.4 percent.

As a result, there's a 0.4 percent chance of getting a negative cash balance after six months.

For 12 months distribution:

The Probability of negative cash position is calculated as follows:

[tex]z = \dfrac{x-\mu}{\sigma} \\ \\ z = \dfrac{0-3.2}{1.3856} \\ \\ z = -2.3094[/tex]

Using the standard distribution tables,

N(-2.31) equals 0.0104 or 1.04 percent.

As a result, there's a 1.04 percent chance of getting a negative cash balance after 1 year  

c) To determine the time period over which the likelihood of achieving a negative cash condition is highest, it's necessary to examine the z-score more closely. Essentially, the z-score measures the difference between a given value(x) and the mean of all potential values [tex](\mu)[/tex], expressed in terms of the total set's standard deviation [tex](\sigma)[/tex]

This suggests that the higher the z-score, the greater the difference occurring between x and [tex]\mu[/tex], and thus the likelihood of receiving x is minimal. As a result, the best chance of finding a certain value is when the z-score is the lowest.

To do so, calculate the derivative of the z-score in relation to the time interval. The point where the derivative is equivalent to zero is where the z-scores are at their lowest.

The first step is to go over the z-score formula in more detail, as seen below.;

[tex]z = \dfrac{x-\mu}{\sigma} \\ \\ z = \dfrac{0-(initial \ value + \alpha T)}{b \sqrt{T}} \\ \\ z = \dfrac{-initial \ value }{b\sqrt{T}}-\dfrac{a \sqrt{T}}{b} \\ \\[/tex]

Now, compute the derivative of this equation with respect to T as follows:

[tex]\dfrac{dz}{dT}= \dfrac{initial \ value \times T^{-\dfrac{3}{2}}}{2b} - \dfrac{aT^{-\dfrac{1}{2}}}{2b}[/tex]

Now, figure out the value of T at which this derivative is equal to zero by substituting all values as follows:

[tex]0 = \dfrac{2.0 \times T^{-\dfrac{3}{2}}}{2\times 0.4}- \dfrac{0.1 \times T^{-\dfrac{1}{2}}}{2 \times 0.4} \\ \\ \\ 0.1 \times T^{-\dfrac{1}{2}}= 2.0 \times T^{-\dfrac{3}{2}} \\ \\ \\T = \dfrac{2}{0.1} \\ \\ \\ T = 20[/tex]

As a result, the time period in which achieving a negative cash condition is = 20 months.

The mathematics faculty at a college consists of 7 ​professors, 4 associate​ professors, 4 assistant​ professors, and 6 instructors. If one faculty member is randomly​ selected, find the probability of choosing a professor or an instructor.

answer (10-4)x4+8 and i will give u brainlyest and 100pts :>>

HELP ASAP!!!!!. A spinner has three sections colored green, purple, and pink. If the spinner is spun twice, how many branches would the tree diagram that represents this situation have? 6 5 9 12

Select all numbers that have an absolute value of 6.3.

Complete the square to find
the vertex of this parabola.
x2 - 8x - y - 4 = 0
([?], [ ]

The ratio of red cars to blue cars in a parking lot was 5 : 3. If there were 40 red cars, how many blue cars were there?

there are 6 rows of chairs 4 rows are empty

Tom works at an aquarium shop cleaning reef tanks. His job is to drain the tank. How much water is initally in
the tank before he starts to drain it?
Time (minutes)
Amount of Water
1
150
3
100
6
25

At a drop-in mathematics tutoring center, each teacher sees 4 to 8 students per hour. The probability that a tutor sees 4 students in an hour is 0.12;5 students, 0.13; 6 students, 0.30; and 7 students, 0.33. Let be the number of students per hour of a randomly selected teacher.

i need help with this!

A rectangle has a side length of 9 centimeters and 6 centimeters. What is the area of the rectangle?

Find the length of the third side. If necessary, round to the nearest tenth.
12
5

Marie found that 4 bags of leaves weigh 6 pounds and 8 bags of leaves weigh 12 pounds. She made a graph to help her find the weight of other numbers of bags. On a coordinate plane, a line is drawn through points (4, 6) and (8, 12). Which ordered pair follows the pattern in the graph? (2, 4) (3, 4) (6, 8) (6, 9)

please help me fast thanks​

geometry math help plz answer all questions

VERY EASY, WILL GIVE 50 POINTS FOR CORRECT ANSWER ASAP AND WILL GIVE BRAINLIEST.

A jacket was originally $284. It is discounted 20%, but doesn’t sell, so it is discounted again by 20% off the sale price. Finally, after sitting in the store for 6 months, it is marked as clearance, at 50% of the lowest sale price. What is the clearance price of the jacket?

how do you tell someone you like them if you get nervous around them?
57+ points if you answer the question

Suppose Huang invests $720 into an annuity every month for five years. At the end of five years, the account has a balance
of $50,000. How much of this balance is interest?
Round your answer to the nearest dollar.
Do NOT round until you have calculated the final answer.

-3/8*2/5 + 2/5 * 5/8

Raquel's Brownies
Raquel has 4 brownies. She wants to give an equal share to each
of her friends. How would Raquel share these brownies equally among herself
and 8 friends? How much of the brownies will each person get? Use words,
pictures, and an equation to explain your thinking.
Estimate

student were asked to choose their favorite food and 5/6 chose either pizza or tacos if 2/6 of the students chose pizza what fraction chose tacos

While standing on a pier, Lynette saw a balloon floating 20 feet above her and a fish swimming 4 feet below the surface of the water.

Let 0 represent the water surface, and let the positive side of 0 represent position above the surface. Which number sentence correctly compares the distances of the balloon and the fish to the surface of the water?
A. |-4| < |20|
B. |-4| > |20|
C. |4| < |20|
D. |4| > |-20|

The Northwood Clock Tower measures 193 meters tall. Dante is creating a scale model of this building using the scale 75 meters : 1 meter.

To the nearest tenth of a meter, what will be the height of the scale model?

PLZ HELP I WILL MAKE YOU BRAINLIEST NO FAKE LINKS

Cell phone Plan A costs$70 per month and comes with a free$500 phone. Cell phone Plan Bcosts$50 per month but does not come with a phone. If you buy the$500 phone and choose Plan B, how many months is ituntil your cost is the same as Plan A's?

Lily has 3 dog toys that are red. One fifth of all her dog toys are red. How many dog toys does Lily have? Complete the explanation. Please I need help!!!!
​

if d is 55 degrees then what is F and E degrees. Explain how you know​

Factor out the greatest common monomial factor from polynomial. Show your work or explain your reasoning

F(x)=0.5^x what is the domain and range