1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
liberstina [14]
3 years ago
15

Support requests arrive at a software company at the rate of 1 every 30 minutes. Assume that the requests arrive as events in a

Poisson process.
a) What is the probability that the number of requests in an hour is between 2 and 4 inclusive? Give your answer to four decimal places.

b) What is the expected number of requests in a 10 hour work day? Give an exact answer.

c) What is the probability that the number of requests in a 10 hour work day is between 20 and 24 inclusive? Give your answer to four decimal places.

d) What is the standard deviation of the number of requests in a 10 hour work day? Give your answer to four decimal places.
Mathematics
1 answer:
rewona [7]3 years ago
8 0

Answer:

a. 0.5413

b. 20

c. 0.3724

d. 4.4721

Step-by-step explanation:

Solution:-

- We will start by defining a random variable X.

           

                     X : The number of support requests arrived

- The event defined by the random variable ( X ) is assumed to follow Poisson distribution. This means the number of request in two distinct time intervals are independent from one another. Also the probability of success is linear within a time interval.

- The time interval is basically the time required for a poisson event to occur. Consequently, each distributions is defined by its parameter(s).

- Poisson distribution is defined by " Rate at which the event occurs " - ( λ ). So in our case the rate at which a support request arrives in a defined time interval. We define our distributions as follows:

                                   X ~ Po ( λ )

                                 

Where,                        λ = 1 / 30 mins

Hence,

                                   X ~ Po ( 1/30 )

a)

- We see that the time interval for events has been expanded from 30 minutes to 1 hour. However, the rate ( λ ) is given per 30 mins. In such cases we utilize the second property of Poisson distribution i.e the probability of occurrence is proportional within a time interval. Then we scale the given rate to a larger time interval as follows:

                                   λ* =  \frac{1}{\frac{1}{2} hr} = \frac{2}{1hr}

- We redefine our distribution as follows:

                                   X ~ Po ( 2/1 hr )

- Next we utilize the probability density function for poisson process and accumulate the probability for 2 to 4 request in an hour.

                           P ( X = x ) = \frac{e^-^l^a^m^b^d^a . lambda^x}{x!}

- The required probability is:

                   P ( 2 \leq X \leq 4 ) = P ( X = 2 ) + P ( X = 3 ) + P ( X = 4 )\\\\P ( 2 \leq X \leq 4 ) = \frac{e^-^2 . 2^2}{2!} +  \frac{e^-^2 . 2^3}{3!} + \frac{e^-^2 . 2^4}{4!}\\\\P ( 2 \leq X \leq 4 ) = 0.27067 + 0.18044 + 0.09022\\\\P ( 2 \leq X \leq 4 ) = 0.5413            Answer    

b)

We will repeat the process we did in the previous part and scale the poisson parameter ( λ ) to a 10 hour work interval as follows:

                               λ* = \frac{2}{1 hr} * \frac{10}{10} = \frac{20}{10 hr}

- The expected value of the poisson distribution is given as:

                             E ( X ) = λ

Hence,

                            E ( X ) = 20  (10 hour work day)    .... Answer

c)

- We redefine our distribution as follows:

                                   X ~ Po ( 20/10 hr )

- Next we utilize the probability density function for poisson process and accumulate the probability for 20 to 24 request in an 10 hour work day.

                           P ( X = x ) = \frac{e^-^l^a^m^b^d^a . lambda^x}{x!}

- The required probability is:

                   P ( 20 \leq X \leq 24 ) = P ( X = 20 ) + P ( X = 21 ) + P ( X = 22 )+P ( X = 23 ) + P ( X = 24 )\\\\P ( 20 \leq X \leq 24 ) = \frac{e^-^2^0 . 20^2^0}{20!} +  \frac{e^-^2^0 . 20^2^1}{21!} + \frac{e^-^2^0 . 20^2^2}{22!} + \frac{e^-^2^0 . 20^2^3}{23!} + \frac{e^-^2^0 . 20^2^4}{24!} \\\\P ( 20 \leq X \leq 24 ) = 0.0883 +0.08460 +0.07691 +0.06688+0.05573\\\\P ( 20 \leq X \leq 24 ) = 0.3724            Answer  

c)

The standard deviation of the poisson process is determined from the application of Poisson Limit theorem. I.e Normal approximation of Poisson distribution. The results are:

                                σ = √λ

                                σ = √20

                                σ = 4.4721 ... Answer

You might be interested in
An outlier always affects the mean.<br><br> True<br> False,<br> Please help ASAP
alexandr1967 [171]
True if u have high or very low outlier in your data set it is generally preferred to use the median    
4 0
3 years ago
PLEASE ANSWER THIS ASAP !!!
Inga [223]

Answer:

Step-by-step explanation:

Finding the answer to the second box is easy. Just look at where the line hits the y axis. That point is (0,5). Put a 5 in the second box.

-2

Now pick two points How about (0,5) and (2,1)

Givens

y2 = 5

y1 = 1

x2 = 0

x1 = 2

Formula

m = (y2 - y1) / (x2 - x1)

m = (5 - 1)/(0 - 2)

m = 4 / - 2

m = - 2

Answer

So the first box contains - 2

7 0
2 years ago
3x + 1 − 4x3 + 6x6 −2x2 standard form
Molodets [167]
<span>6x6 - 4x3 - 2x2 + 3x + 1</span>
7 0
3 years ago
The $1 coin depicts Sacagawea and her infant son. The diameter of the coin is 26.5 mm, and the thickness is 2.00 mm. Find the vo
Contact [7]

Answer

Find the volume of the coin is cubic millimeters.

To prove

Formula

Volume\ of\ cylinder\ = \pi r^{2} h

Where r is the radius and h is the height .

As given

The $1 coin depicts Sacagawea and her infant son.

The diameter of the coin is 26.5 mm, and the thickness is 2.00 mm.

Radius = \frac{Diameter}{2}

Radius = \frac{26.5}{2}

Radius = 13.25 mm

\pi = 3.14

Put in the formula

Volume  of  coin  = 3.14 × 13.25 × 13.25 × 2.00

                            = 1102.53 mm³ (approx)

Therefore the volume of the coin is 1102.53 mm³ .

7 0
3 years ago
The graph of h(x) is shown. Graph of h of x that begins in quadrant two and decreases rapidly following the vertical line which
fgiga [73]

Answer:

<u>x-intercept</u>

The point at which the curve <u>crosses the x-axis</u>, so when y = 0.

From inspection of the graph, the curve appears to cross the x-axis when x = -4, so the x-intercept is (-4, 0)

<u>y-intercept</u>

The point at which the curve <u>crosses the y-axis</u>, so when x = 0.

From inspection of the graph, the curve appears to cross the y-axis when y = -1, so the y-intercept is (0, -1)

<u>Asymptote</u>

A line which the curve gets <u>infinitely close</u> to, but <u>never touches</u>.

From inspection of the graph, the curve appears to get infinitely close to but never touches the vertical line at x = -5, so the vertical asymptote is x = -5

(Please note:  we cannot be sure that there is a horizontal asymptote at y = -2 without knowing the equation of the graph, or seeing a larger portion of the graph).

5 0
2 years ago
Read 2 more answers
Other questions:
  • If 5 adult movie tickets cost $25, how much do 11 tickets cost?
    8·1 answer
  • for the breakfast buffet, mr. walker must cut and equally divide 12 loaves of bread over 7 platters. how many loaves of bread ar
    7·2 answers
  • The line graph shows the average daily cost, rounded to the nearest 10 cents, that a homeowner paid for electricity each month o
    8·2 answers
  • Calculate s26 for the arithmetic sequence in which a14=4.1 and the common difference is d=1.7
    6·1 answer
  • I need help on this I don’t understand how to do this
    14·1 answer
  • Minh is 16. His parents are both the same
    7·2 answers
  • A ___________is a group of places with shared physical or human characteristics.
    5·1 answer
  • E1xpress the given statement in the form of expression.
    13·1 answer
  • List 5 fractions that are between 1/3 and 4/5
    15·1 answer
  • Gas mileage is the number of miles you can drive on a a gallon of gasoline. A test of a new car results in 510 miles on 10 gallo
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!