Question

A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean μ = 1000. (i) Use this model to find the probability that a bulb fails within the first 400 hours. (Round your answer to three decimal places.)

Answer 1

P(0≤T≤400)=0.3296

Explanation:

we know probability density function is

`f(x;\lambda)= \left\{\begin{matrix} \lambda e^(-\lambda x)& x\geq 0 \\0 & x< 0\end{matrix}\right.`

`\lambda=(1)/(\mu)`

`\lambda=(1)/(1000)`

probability that bulb will fail within first 400 hours

P(0≤T≤400)=
`\int\limits^(400)_0 {f(x)} \, dx`

P(0≤T≤400)=
`\int\limits^(400)_0 {\frac{e^{-(t)/(1000)}}{1000}} \, dx`

=
`(1000)^(-1)[\frac{e^{(-t)/(1000)}}{-(1000^(-1))}]^(400)_0`

=
`-[e^{(-400)/(1000)}-e^0]`

=1-0.67032 = 0.3296

so about 32.96 % bulb will fail within 400 hrs.