Question

The lifetime of a radio is exponentially distributed with a mean of ten years. If Jones buys a ten-year-old radio, what is the probability that it will be working after an additional ten years?

Short answer

The probability that Jones' 10-year-old radio will work after an additional 10 years is approximately 36.79%.

Step-by-step solution

Understand the Exponential Distribution

The exponential distribution is a continuous probability distribution with a probability density function given by: \(f(x;\lambda) = \lambda e^{-\lambda x}\) where x is the time (in this case the age of the radio) and λ is the hazard rate parameter. The mean of the exponential distribution is equal to \(\frac{1}{\lambda}\), so given that the mean lifetime is 10 years, we can calculate λ: \(10 = \frac{1}{\lambda} \)

Calculate the hazard rate parameter λ

Solve for λ using the mean given in the problem: \(\lambda = \frac{1}{10}\)

Understand the survival function

The survival function, S(x), is the probability that the radio survives (is still working) beyond age x. The survival function is the complement of the cumulative distribution function (the probability that the radio has already failed before age x). For an exponential distribution, the survival function is given by: \(S(x) = 1 - F(x)\) Where S(x) is the survival function and F(x) is the cumulative distribution function. For an exponential distribution, the cumulative distribution function F(x) is given by: \(F(x;\lambda) = 1 - e^{-\lambda x}\)

Calculate survival probability

We are interested in the survival probability after an additional 10 years (x=10). Therefore, using the exponential distribution survival function, we can calculate the probability: \(S(10) = 1 - F(10)\) \(S(10) = 1 - (1 - e^{-\frac{1}{10} \times 10})\) \(S(10) = 1 - (1 - e^{-1})\) \(S(10) = e^{-1}\)

Find the probability

Calculate the probability: \(S(10) \approx 0.3679\) The probability that Jones' 10-year-old radio will work after an additional 10 years is approximately 36.79%.