Question

ppose the lifetime (in months) of an engine, working under hazardous conditions, has an exponential distribution with mean of 5 months. an engine is termed successful if its lifetime exceeds 10 months. in a sample of 10 engines, what is the probability of at least 2 successful engines?

Answer 1

To solve this problem, we use the concept of the exponential distribution and the cumulative distribution function. The probability of at least 2 successful engines can be calculated by finding the complement of the probability of less than 2 successful engines.

Step-by-step explanation:

To solve this problem, we need to use the concept of the exponential distribution. The exponential distribution is often used to model the time to failure of a system, and it has a probability density function given by f(x) = λe^(-λx), where λ is the rate parameter.

In this case, the mean of the exponential distribution is given as 5 months, which corresponds to the rate parameter λ = 1/5. We are interested in finding the probability of at least 2 successful engines, which means the lifetime of the engine exceeds 10 months.

To calculate this probability, we can use the cumulative distribution function (CDF) of the exponential distribution, which gives the probability that the lifetime of an engine is less than or equal to a given value. The CDF of the exponential distribution is given by F(x) = 1 - e^(-λx).

The probability of at least 2 successful engines can be calculated as the complement of the probability of less than 2 successful engines:

1. Calculate the probability of less than 2 successful engines: P(X <= 10) = 1 - e^(-λ * 10)
2. Calculate the complement probability: P(at least 2 successful engines) = 1 - P(X <= 10)

Substituting the value of λ = 1/5, we can calculate the probability using the given formula.