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Statistics / Applied Statistics and Probability for Engineers 6 / Chapter 4.12 / Problem 224SE

Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger

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An article in Journal of Theoretical Biology [“Computer

Chapter 4, Problem 224SE

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QUESTION:

An article in Journal of Theoretical Biology [“Computer Model of Growth Cone Behavior and Neuronal Morphogenesis” (1995, Vol. 174(4), pp. 381–389)] developed a model for neuronal morphogenesis in which neuronal growth cones have a significant function in the development of the nervous system. This model assumes that the time interval between filopodium formation (a process in growth cone behavior) is exponentially distributed with a mean of 6 time units. Determine the following:

(a) Probability formation requires more than nine time units

(b) Probability formation occurs within six to seven time units

(c) Formation time exceeded with probability 0.9

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QUESTION:

An article in Journal of Theoretical Biology [“Computer Model of Growth Cone Behavior and Neuronal Morphogenesis” (1995, Vol. 174(4), pp. 381–389)] developed a model for neuronal morphogenesis in which neuronal growth cones have a significant function in the development of the nervous system. This model assumes that the time interval between filopodium formation (a process in growth cone behavior) is exponentially distributed with a mean of 6 time units. Determine the following:

(a) Probability formation requires more than nine time units

(b) Probability formation occurs within six to seven time units

(c) Formation time exceeded with probability 0.9

ANSWER:

Solution :

Step 1 of 3:

Given the time interval between filopodium formation is exponentially distributed with a mean of 6 time units.

So $\mu{}=6$ .

The formula for the exponential probability distribution is

P( $x\leq{}a$ ) = 1- ${e}^{\frac{-a}{\mu{}}}$

P( $x\geq{}b$ ) = 1- ${e}^{\frac{-b}{\mu{}}}$

P( $a<x<b$ ) = ${e}^{\frac{-a}{\mu{}}}-{e}^{\frac{-b}{\mu{}}}$

Our goal is:

a). We need to find the probability formation requires more than 9 time unit.

b). We need to find the probability of within six to seven time units.

c). We need to find the formation time exceeded with probability 0.9.

a). The probability formation requires more than 9 time units is

We know that $\mu{}=6$ and b=6.

P( $x\geq{}9$ ) = ${e}^{\frac{-9}{6}}$

P( $x\geq{}9$ ) = ${e}^{-1.5}$

P( $x\geq{}9$ ) = 0.2231

Therefore, the probability formation requires more than 9 time units is 0.2231.

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