A new iPhone user receives an average of 4.2 phone calls per hour. The calls occur randomly in time so if X = the number of calls received in an hour, then we can assume that X has a Poisson distribution with parameter λ = 4.2. Let Y = the number of phone calls received in a day. Assume all necessary independence in the following.
a. Y also has a Poisson distribution. What is the λ for Y?

b. What is the standard deviation of X?

c. What is the standard deviation of Y?

d. What is the probability that no calls are received in an hour?

e. What is the probability that more than 120 calls are received in a day?

f. What is the probability that Y is between 90 and 110 inclusive? (i.e. 90 ≤ Y≤110)

g. What is the variance of square root of 2Y?

h. Assume this distribution holds for all hours and days and that one call is independent from another. What is the probability that 120 or fewer calls are received for every day of this week? (Assume appropriate independence

Add any comments below.

Answers

Answer 1

Part A:

Given that Y is the number of phone calls received in a day and that a new iPhone user receives an average of 4.2 phone calls per hour.

Thus, the average number of phone calls received in a day is 4.2 * 24 = 100.8

Therefore, the λ for Y is 100.8

Part B:

The standard deviation of a Poisson distribution is given by

[tex]\sigma=\sqrt{\lambda}[/tex]

Thus, the standard deviation of X is given by

[tex]\sigma=\sqrt{4.2}=2.05[/tex]

Therefore, the standard deviation of X is 2.05.

Part C:

The standard deviation of Y is given by

[tex]\sigma=\sqrt{100.8}=10.04[/tex]

Therefore, the standard deviation of X is 10.04.

Part D:

The probability of a Poisson distribution is given by

[tex]P(X=x)=e^{-\lambda} \frac{\lambda^x}{x!} [/tex]

where, x is the number of occurence of the event in the given period.

Thus, the probability that no calls are received in an hour is given by

[tex]P(X=0)=e^{-4.2} \frac{(4.2)^0}{0!} =0.015[/tex]

Therefore, the probability that no calls are received in an hour is 0.015

Part E:

We approximate the Poisson distribution using Normal distribution by noting that for a Poisson distribution

[tex]\mu=\lambda \\ \\ \sigma^2=\lambda \\ \\ \sigma=\sqrt{\lambda}[/tex]

The probability of a normal distribution is given by

[tex]P(X\ \textless \ x)=P\left(z\ \textless \ \frac{x-\mu}{\sigma} \right)[/tex]

Thus, the probability that more than 120 calls are received in a day is given by

[tex]P(X\ \textgreater \ 120)=1-P(X\leq120) \\ \\ =1-P\left(z\leq \frac{120-100.8}{10.04} \right)=1-P(z\leq1.912) \\ \\ =1-0.97208=0.0279[/tex]

Therefore, the probability that more than 120 calls are received in a day is 0.0279

Part F:

We also approximate this using Normal distribution

The probability of a normal distribution between two value (a, b) is given by

[tex]P(a\ \textless \ X\ \textless \ b)=P\left(z\ \textless \ \frac{b-\mu}{\sigma} \right)-P\left(z\ \textless \ \frac{a-\mu}{\sigma} \right)[/tex]

Thus, the probability that Y is between 90 and 110 inclusive is given by

[tex]P(90\leq X\leq 110)=P\left(z\leq \ \frac{110-100.8}{10.04} \right)-P\left(z\leq \frac{90-100.8}{10.04} \right) \\ \\ =P(z\leq0.916)-P(-1.076)=0.82025-0.14103=0.67922[/tex]

Therefore, the probability that Y is between 90 and 110 is 0.67922

Part G:

The variance of aY is given by

[tex]Var(aY)=a^2Var(Y)[/tex]

Thus, the variance of [tex] \sqrt{2} Y[/tex] is given by

[tex]Var( \sqrt{2} Y)=( \sqrt{2} )^2Var(Y)=2(100.8)=201.6[/tex]

Therefore, the variance of [tex] \sqrt{2} Y[/tex] is 201.6

Part H:

We approximate this using Normal distribution

The probability of a normal distribution is given by

[tex]P(X\ \textless \ x)=P\left(z\ \textless \ \frac{x-\mu}{\sigma} \right)[/tex]

Thus, the probability that 120 or fewer calls are received in a day is given by

[tex]P(X\leq120)=P\left(z\leq \frac{120-100.8}{10.04} \right) \\ \\ =P(z\leq1.912)=0.97208[/tex]

Since, this distribution holds for all hours and days and one call is independent from another. Assuming that there are seven days in a week.

The probability that 120 or fewer calls are received for every day of this week is given by

[tex](0.97208)^7=0.8202[/tex]

Therefore, the probability that 120 or fewer calls are received for every day of this week is 0.8202

Related Questions

Please solve-y(2+4y)

Answers

To solve this, you distribute the -y to both terms, 2 and 4y.
It will look something like this: (2*-y)+(4y*-y)
When you simplify, look for any like terms that can be combined, and if none, then leave it as it is.
Your final answer will be: -2y-4y^2

When performing today's experiment, your lab partner decides to add all the reagents into one beaker instead of two separate beakers. after adding everything together and waiting a significant amount of time, you see no reaction occur. what was the problem?

Answers

The problem can be an incorrect sequence of reagents added, requirement of heat or proportion of relative reactants when all the reagents simultaneously added into the beaker.

Further explanation:

The reaction sometimes occurs in a series of elementary reaction which can be described in the reaction mechanism of that particular reaction. So when we added the incorrect sequence of reagents then the reaction may not happen as the way we want.

Some factors which can affect the reaction are as follows:

1. The ratio amount of given reagent may not be correct.

2. Some of the reactions have very slow rate of reactions and thus they required heat to increase the rate of reaction.

3. Sometimes the reagent which is important for the first step can undergo a side reaction with the reagent which should not be there otherwise.

4. That mechanism of reaction may be required light since chemoluminescence reactions depend on excitation of atoms.

5. Sometimes reactants should be mix in dark which can lose their activity in the light.

Learn more:

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Answer details:

Grade: Senior School

Subject: Chemistry

Chapter: Keys to study chemistry

Keywords: Performing experiment, lab partner, add all the reagents into one beaker, two separate beakers, and no reaction occur.

what's the closest number to 27.8 * 9.8

Answers

280...round each number to the nearest whole number (27.8=28 and 9.8=10) then multiply and you will get 280

27.8 * 9.8 = 272.44

Closest whole number = 272.

Complete the sentence using one of the words below in 10/6 the 10 is the

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The 10 Is The Numerator

the ratio to strawberry to banana batter is 2:3. We need 30 cups of batter total. How much banana batter do we need to make?

Answers

The answer is eighteen

what is -(-2)??????

Answers

-(-2) is 2

Because -1 and -2 make a +2

- (-2) = 2

two negatives = positive

one positive and one negative = negative

two positives = positive

hope this helps

In this henrik's class, there are 10 boys and 12 girls. What is the ratio of boys to girls? Express your answer as a fraction in simplest form.

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5/6 is the simplest ratio

Have an average weight of 16 ounces and a standard deviation of 0.2 ounces. the weights of the sugar bags are normally distributed. what is the probability that 16 randomly selected packages will have a weight in excess of 16.075 ounces?

Answers

Standardised z score = (16.075 - 16) / 0.2 = 0.375

which gives a probability of 0.1462 for selecting a package of weight 16.075

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Probability of picking 16 in excess of 16-075 = 0.3538^16 = 6.0273 * 10^-8

In a football game, the home team gained 8 yards in a running play. After that, the quarterback was sacked for a loss of 8 yards. On the next play, the quarterback threw a 25-yard pass for a touchdown. What was the final yardage gain on the play? Justify your answer.

Answers

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Here again is the information about the prescription drug: there is a 10% chance of experiencing a sore throat (s) and a 30% chance of experiencing a fever (f). the information also states that there is a 6% chance of experiencing both side effects. what is the probability of experiencing neither of the side effects?

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youve got a 94% chance of not experiencing it but a 90% chance of not experiencing a sore throat and a 70% cahnce of not experiencing a fever

A map of a public park shows a circular pond. There is a bridge along a diameter of the pond that is 0.25 mi long. You walk across the bridge, while your friend walks halfway around the pond to meet you at the other side of the bridge. How much farther does your friend walk?

Answers

The circumference of a circle is pi times diameter, so half-way around the pond is (1/2)pi times diameter.

[tex]\frac{\pi}{2} = 1.57[/tex]

The difference between walking around the pond and walking across the bridge is: 1.57d - d = 0.57d

Plug in diameter of 0.25

0.25*0.57 = 0.14 miles

Final answer:

Your friend walks approximately 0.3925 miles farther than you along the curved path of the pond.

Explanation:

To find out how much farther your friend walks, we need to compare the distance along the bridge to the distance halfway around the pond. The bridge is a straight line across the pond, which is the same as the diameter of the pond. The diameter is 0.25 miles long. Your friend walks halfway around the pond, which is equivalent to walking half the circumference of the pond. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. Since the bridge is the diameter, it is equal to 2 times the radius, or 2r. So the distance your friend walks is half the circumference or (1/2)(2πr) = πr. Since the diameter is 0.25 miles, the radius is half of that, or 0.125 miles. Plugging this into the formula, we find that your friend walks π(0.125) miles. To find a numerical approximation, we can use the fact that π is approximately 3.14, so your friend walks 3.14(0.125) miles. This simplifies to approximately 0.3925 miles. Therefore, your friend walks approximately 0.3925 miles farther than you along the curved path of the pond.

Use addition to simplify the expression 6/x+6/3x . show all the steps necessary to write the expression with a common denominator and to simplify the expression

Answers

Common Denominator is 3x

we write 6/x as 18/3x So we have

18/3x + 6 /3x = 24/3x

simplifyiby dividing numerator and denominator by 3:-

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How many times dose 63 go into 272? THANK YOU SO MUCH FOR ANSWERING,IF YOU DID!!!!!!!

Answers

Let's write this problem as an expression: 272 ÷ 63
Now, it is just division, and we can divide the two numbers to get 4.317...
However, the question only asks for how many times 63 goes into 272, so the answer would be that 63 goes into 272 four times. Hope this helps and have an awesome day!

Which expression makes the equation true? X^8 = ( )^2

Answers

x^8 = ()^2

x^8 = (x^4)^2

x^8 = x^8

inside the parethesis is x^4

hope this helps

(a^m)^n = a^(mn)

x^8 = (x^4)^2 = x^(4 * 2) = x^8

Answer: (x^4)^2

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Answers

87.36
This can be rounded to equal 87.4.

Hope this helps! :)

85 would be a good estimate.
the estimate is close to the real answer which is 87.36.
remember, estimates are not exact answers but they are close guesses.

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Answers

-5(-2) - 6 = 10 - 6 = 4 (answer)

4 is the answer because

-5g-6=
-5(-2)-6=
10-6=
4

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Answers

10,000/100=100
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Answers

Final answer:

There are 558 chairs in total.

Explanation:

To find the total number of chairs, you need to multiply the number of rows by the number of chairs in each row. In this case, there are 62 rows and 9 chairs in each row. So, the total number of chairs would be 62 x 9 = 558.

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Answers

True, Integers mean "whole", and it cannot be a fraction or decimal

hope this helps

true , hope this helps :)

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Answers

For graph 1, the y-intercept is (0,1) and the slope is 1.
For graph 2, the y-intercept is (0,4) and the slope is -1.
For graph 3, the y-intercept is (0,-4) and the slope is -2.
For graph 4, the y-intercept is (0,-3) and the slope is 3/2.
For graph 5, the y-intercept is (0,4) and the slope is 2.
For graph 6, the y-intercept is (0,3) and the slope is -3/2.

(By the way, since the graphs aren't labeled as numbers, I went from left to right, and then down, and then left to right again).

Now, here's the explanation:

Solving for y-intercept is easier than it sounds. All you need to do is mark the point where the line crosses the y-axis (vertical axis).

Solving for slope is just rise/run. For example, in graph one, the points on the line went up one and then to the right one.

So, 1/1 = 1.

When a slope moves down or to the left, it will be negative. However, that's only if the rise OR run is down/left. If both are negative, the two will cancel each other out to make a positive.

Hope this helped!!

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Answers

You need to find a number that 12 and 20 share so you go through all the number starting with 1 and go until you can find one so I'll use the numbers 2 and 4. Gives you two options.
12/2=6
20/2=10
6:10

12/4=3
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3:5

Anisa is cooling water in her fridge.The waters temperature starts at 85 degrees and she cools it 3.5 degrees every minute for 12 minutes.Create a number sentence to model this scenario.

Answers

The temperature starts at 85 degrees.
t = 85

After 1 minute, it loses 3.5 degrees.
t = 85 - 3.5

After 2 minutes, it loses another 3.5 degrees.
t = 85 - 3.5 * 2

After 3 minutes, it loses another 3.5 degrees.
t = 85 - 3.5 * 3

etc.

After 12 minutes, it loses another 3.5 times 12 degrees.
t = 85 - 3.5 * 12

Two pounds of bananas cost $1.16 what would be the cost of 5pounds s of bananas

Answers

If 2 lb of bananas cost $1.16, then each pound costs $0.58.

0.58 * 5 = 2.9

So 5 lb of bananas will cost $2.90.

Hope this helps!

The cost of 5 lbs of bananas would be $2.90.

Shelly sells 5 bone shaped treats for $3.50 how much should she charge to a pack of 12 treats which proportion is needed to solve the problem

Answers

Hola en ese caso se realiza una regla de tres.
si shelly cobra 5 golosinas por 3.50 dolares, 12 golosinas a cuanto equivaldran.

(12x3.50)/ 5 = 8.4 dolares deberia cobrar

Answer

5/3.5and 12/x or d

Car 6 truck 10 suv 14 minivan 15 at this rate how many trucks would pass Luann if 90 vehicles passed her?

Answers

I think that there would be 20 trucks because 6 + 10 + 14 + 15 = 45, 6 + 10 + 14 + 15 = 45, 45 + 45 = 90

How many standard deviations tend to fall on either side of the mean in a distribution?

Answers

This percentage of scores fall within one standard deviation of the mean. 68% This percentage of scores fall on either side of the distribution: 50%

Avery collect cans for recycling when he has 1,000 cans the recycling Center will pick them up from the house Avery has 80 bags with about 40 cans in each bag should he call the recycling center to arrange to pick up explain how you know

Answers

He should call. If he has 80 bags and each of them have 40 cans, you would do 80x40, which is 3200. He has more than enough cans for the recycling center to pick up

What property of addition is shown in the following number sentence? Explain 7+(3+5)=(7+3)+5

Answers

The property: Associative Property of Addition. 7 + (3 + 5) = (7 + 3) + 5
this is the associative property....it has to do with grouping numbers..u can add the numbers no matter how the numbers are grouped and u will get the same answer

The required property is associative property.

Given that,
To determine a property of addition is shown in the following number sentence 7+(3+5)=(7+3)+5.

What is simplification?

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
7 + (3 + 5) = (7 + 3) + 5
Here, accoding to the PEMDAS rule, in mathematical string solution of the parenthesis is on priority. But we can see here on the left we add first 3 and 5 then the product adds to 7 and on the right, we add first 7 and 3 then the product adds to 5 gives the same result, this property of addition of number is called associative property.
As
7 + (3 + 5) = (7 + 3) + 5
7 + 8 = 10 + 5
15 = 15

Thus, the required property is associative property.

Learn more about simplification here: https://brainly.com/question/12501526

#SPJ2

Plot the x- and y-intercepts to graph the equation. y=−12x−3

Answers

x-intercept: (1/4) ,0
y-intercept: (0, -3)

Keywords

x-intercept, y-intercept, linear equation, graphing tool

we know that

The y-intercept is the value of y when the value of x is equal to zero

The x-intercept is the value of x when the value of y is equal to zero

In this problem we have

[tex]y=-12x-3[/tex]

Step 1

Find the x-intercept

For [tex]y=0[/tex]

find the value of x

[tex]0=-12x-3[/tex]

[tex]12x=-3[/tex]

[tex]x=-0.25[/tex]

The x-intercept is the point [tex](-0.25,0)[/tex]

Step 2

Find the y-intercept

For [tex]x=0[/tex]

find the value of y

[tex]y=-12*0-3[/tex]

[tex]y=-3[/tex]

The y-intercept is the point [tex](0,-3)[/tex]

Step 3

With the point [tex](-0.25,0)[/tex] and the point [tex](0,-3)[/tex]

Plot the linear equation

Using a graphing tool

see the attached figure

therefore

the answer in the attached figure

Harry rides his bike 25 Start Fraction 3 over 8 End Fraction miles. Karen rides her bike 25two-fifths miles. Who rides farther?

Answers

I believe Harry rode the farthest.

Hope this helps!

Final answer:

After converting their distances to decimals, Karen rides farther than Harry with a distance of 25.4 miles compared to Harry's 25.375 miles.

Explanation:

The question is regarding who rides farther, Harry or Karen, given their distances in different fractional forms. Harry rides his bike 25 3/8 miles, and Karen rides 25 two-fifths miles. To compare, we must convert their distances to similar fractional forms or decimals. Converting three-eighths to a decimal gives us 0.375, and converting two-fifths to a decimal yields 0.4. Therefore, Harry rides 25.375 miles, and Karen rides 25.4 miles.

Through comparison, it is clear that Karen rides farther than Harry. Karen's total distance of 25.4 miles exceeds Harry's total distance of 25.375 miles, albeit by a slight margin.

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