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The time until a chemical reaction is complete (in milliseconds) is approximated by the cumulative distribution function: F(x)={

1−e
−0.01x

0


x≥0
x<0

Determine the probability density function of X. What proportion of reactions is complete within 200 milliseconds?

1 Answer

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To determine the probability density function (PDF) of X, we can take the derivative of the cumulative distribution function (CDF) with respect to x. Let's denote the PDF as f(x).

Given that F(x) = 1 - e^(-0.01x) for x ≥ 0 and F(x) = 0 for x < 0, we have:

f(x) = d/dx [F(x)]

For x ≥ 0, applying the derivative to F(x):

f(x) = d/dx [1 - e^(-0.01x)]

= 0 - (-0.01)e^(-0.01x)

= 0.01e^(-0.01x)

For x < 0, the PDF is 0 since the CDF is 0 for negative values.

Therefore, the probability density function of X is given by:

f(x) = {

0.01e^(-0.01x), x ≥ 0,

0, x < 0

}

To find the proportion of reactions that are complete within 200 milliseconds, we need to calculate the probability P(X ≤ 200). This can be obtained by integrating the PDF from 0 to 200:

P(X ≤ 200) = ∫[0, 200] f(x) dx

= ∫[0, 200] 0.01e^(-0.01x) dx

To calculate the integral, we can use integration techniques such as substitution or integration by parts. Using the substitution u = -0.01x, du = -0.01 dx, the limits of integration change as well:

P(X ≤ 200) = ∫[0, 200] 0.01e^(-0.01x) dx

= ∫[0, -2] e^u du (with the new limits of integration)

= [e^u]_|0|^(-2)

= e^(-2) - e^0

= e^(-2) - 1

Hence, the proportion of reactions that are complete within 200 milliseconds is approximately e^(-2) - 1.

User Evgeny Gorb
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