A machine can process two online applications in 60 minutes. Answer the following questions.What would be an appropriate random variable? What would be the exponential-distribution counterpart to the random variable?Identify the mean of x, where x = minutes until the next occurrence and determine the following: P (x ≥ 30.0)P (x ≤ 10)The machine must be disconnected for service for 45 minutes. What is the probability that an online application will be sent while the machine is out of service?

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Description: A machine can process two online applications in 60 minutes. Answer the following questions.What would be an appropriate random variable? What would be the exponential-distribution counterpart to the random variable?Identify the mean of x, where x =

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A machine can process two online applications in 60 minutes. Answer the following questions.

What would be an appropriate random variable? What would be the exponential-distribution counterpart to the random variable?
Identify the mean of x, where x = minutes until the next occurrence and determine the following:
1. P (x ≥ 30.0)
2. P (x ≤ 10)
The machine must be disconnected for service for 45 minutes. What is the probability that an online application will be sent while the machine is out of service?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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Given that a machine processes 2 applications per hour. This process can be modeled as a Poisson process with rate $λ$

(1) Let $λ > 0$ be fixed. The counting process ${N (t), t \in [0, \infty)}$ is called a Poisson process with rates $λ$ if all the following conditions hold:

$N (0) = 0$ ;
$N (t)$ has independent increments;
the number of arrivals in any interval of length t>0 has $P o i s s o n (λ τ)$ distribution.

The exponential distribution counterpart to the Poisson random variable is as follows:

If $N (t)$ is a Poisson process with rate $λ$ , then the inter arrival times $X_{1}$ , $X_{2}$ , $\dots$ are independent and

X i \sim E x p o n e n t i a l (λ), for i = 1, 2, 3, \dots.

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