A reservation service employs six information operators who receive requests for information independently of one another, each according to a Poisson process with rate ???? = 2 per minute. a. What is the probability that during a given 1 min period, the first operator receives no requests? (Round your answer to three decimal places.) b. What is the probability that during a given 1 min period, exactly three of the six operators receive no requests? (Round your answer to five decimal places.)

Answers

Answer 1

Answer:

Answer:

a) 0.135 = 13.5% probability that during a given 1 min period, the first operator receives no requests.

b) 0.03185 = 3.185% probability that during a given 1 min period, exactly three of the six operators receive no requests

Step-by-step explanation:

To solve this question, we need to understand the Poisson distribution and the binomial distribution.

Poisson distribution:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

$P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)$

In which

x is the number of sucesses

e = 2.71828 is the Euler number

$\mu$ is the mean in the given time interval.

Binomial distribution:

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)$

In which $C_(n,x)$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_(n,x) = (n!)/(x!(n-x)!)$

And p is the probability of X happening.

Poisson process with rate 2 per minute

This means that $\mu = 2$

a. What is the probability that during a given 1 min period, the first operator receives no requests?

Single operator, so we use the Poisson distribution.

This is P(X = 0).

$P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)$

$P(X = 0) = (e^(-2)*2^(0))/((0)!) = 0.135$

0.135 = 13.5% probability that during a given 1 min period, the first operator receives no requests.

b. What is the probability that during a given 1 min period, exactly three of the six operators receive no requests?

6 operators, so we use the binomial distribution with $n = 6$

Each operator has a 13.5% probability of receiving no requests during a minute, so $p = 0.135$

This is P(X = 3).

$P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)$

$P(X = 3) = C_(6,3).(0.135)^(3).(0.865)^(3) = 0.03185$

0.03185 = 3.185% probability that during a given 1 min period, exactly three of the six operators receive no requests

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Evaluate the function.
f(x)=-2x^2-7x+6

Find f(−3)

Answers

Answer:

f(-3) = 9

Step-by-step explanation:

Step 1: Define

f(x) = -2x² - 7x + 6

f(-3) = x = -3

Step 2: Substitute and evaluate

f(-3) = -2(-3)² - 7(-3) + 6

f(-3) = -2(9) + 21 + 6

f(-3) = -18 + 21 + 6

f(-3) = 3 + 6

f(-3) = 9

Substituting x = -3 into the function f(x) = -2x^2 - 7x + 6, we get f(-3) = -33. Therefore, the value of the function at x = -3 is -33.

To evaluate the function f(x)=−2x ^2 −7x+6 at x=−3, substitute −3 for x in the function: f(−3)=−2{(−3) }^2 −7(−3)+6

Now, calculate each part of the expression:

(−3) ^2 is 9 because the square of −3 is 9.

−2 times 9 is −18 because −2⋅9 =−18.

−7 times −3 is 21 because −7⋅−3=21.

Now, plug these values back into the expression:

f(−3)=−18−21+6

Finally, add and subtract:

f(−3)=(−18−21)+6=−39+6=−33

So, f(−3)=−33.

The value of the function

f(x)=−2x^ 2 −7x+6 at x=−3 is −33.

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Trick or treaters arrive to your house according to a Poisson process with a constant rate parameter of 20 per hour. Suppose you begin sitting on your front porch, observing these arrivals, at some point in time. Suppose a trick or treater arrived 30 minutes ago, but there have been none since. What is the expected value of interarrival time (in minutes) from the previous trick or treater to the next one

Answers

Answer:

The expected value of interarrival time (in minutes) from the previous trick or treater to the next one is 3 minutes.

Step-by-step explanation:

We have a Poisson process with a constant rate parameter of 20 arrival per hour.

This type of processes are memory-less, meaning that no matter how much time has passed form the last event, the probabilities of an arrival stay the same.

The mean interarrival time can be calculated as the inverse of the mean arrival for the Poisson process.

If the Poisson process has a rate of 20 arrivals per hour, the mean interarrival time is:

$\bar t=1/\lambda=1/20=0.05\,h/arrival=3 \,min/arrival$

Given f(x)=5x+5f(x)=5x+5, find f(-4)f(−4).

Answers

Step-by-step explanation:

f(x) = 5x + 5

f(-4) = 5.(-4) + 5

f(-4) = -20 + 5

f(-4) = -15

A person invested 3,900 in an account growing at a rate allowing the money to double every 7 years. How much money would be in the account after 18 years to the nearest dollar