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Hello, I was going through the question (link below) and did not understand the calculation. It...

Hello, I was going through the question (link below) and did not understand the calculation.
It would be very helpful if you could check the workout and explain the problem to me in simple words.
Thank you
https://www.chegg.com/homework-help/Probability-And-Statistics-for-Engineers-And-Scientists-4th-edition-chapter-5.7-problem-24P-solution-9781111827045

math   Statistics-And-Probability   
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Answer #1

Let us start with one component only. Since failure time has exponential distribution with mean 32 days so X has exponential distribution with mean 32 days. The parameter of X is

The pdf of X is

The probability that a component last longer than 25 days is

----------------

Half of components means 240 /2 = 120

Let Y is the variable shows the number of components out of 240 last longer than 25 days. Here Y has binomial distribution with parameter n=240 and p = 0.4578.

So the probability that at least half of the components last longer than 25 days is

Excel function used for above probability : "=1-BINOMDIST(119,240,0.4578,TRUE)"

Answer: 0.1063

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