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Problem 38 Hard Difficulty

Electronic Device The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without falling for at least five hours. Choose one of the following. Source: Society of Actuaries.
$$
\begin{array}{lllllll}{\text { a. } 0.07} & {\text { b. } 0.29} & {\text { c. } 0.38} & {\text { d. } 0.42} & {\text { e. } 0.57}\end{array}
$$

Answer

option (d) 0.4204

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Calculus 1 / AB

Finite Mathematics and Calculus with Applications

Chapter 18

Probability and Calculus

Section 3

Special Probability Density Functions

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Video Transcript

Okay, so we're told that we have and exponentially distributed raining variable, but we're not told about it in this standard way. We're told the media is equal to four. So, uh, if we remember the formula that the median is equal to Ellen of to over Lambda for an exponentially distributed and unbearable, this becomes nice and easy for us to figure out what Lambda is equal to Lambda is equal to Alan off to over four, and then we know that are variable has a pdf of Allen of to over four e to the negative Allen of to next sober for sense of here. We want a product calculate. Probably the component will work without failing for at least five. So we want the 11 minus probability that it fails within five. So that's gonna be one minus and girl from 0 to 5 of our density function. So it's gonna be one minus. Ah, our density function will integrate up to e to the negative Ln of to X over for evaluated from 50 And so, if we plug in our endpoints, we're going to get one minus one minus e to the negative Ellen, too. Times five divided by. So this is kind of a a lot of junk going on. But if we just go ahead and carefully plug it into a calculator, we will get an answer of approximately. Put 4 to 0 for 48 which rounds 2.42 for our final answer.

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