Question

A type of light bulb is labeled 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. Use this model to find the probability that a bulb fails within the first 300 hours.

Answer 1

`0.2592 \ or \ 25.92\%`

Step-by-step explanation:

The exponential density function is given as

`f(t)=\left \{ {{0} \atop {ce^(ct)}} \right\\0,t<0\\ce^(ct),t\geq 0`

`\mu=(1)/(c)\\c=(1)/(\mu)\\\\=(1)/(1000)=0.001\\\\f(t)=0.001e^(-0.001t)`

To find probability that bulb fails with the first 300hrs, we integrate from o to 300:

`P(0\leq X\leq 300)=\int\limits^(300)_0 {f(t)} \, dt\\\\=\int\limits^(300)_0 {0.001e^(-001t)} \, dt\\ =|-e^(-0.001t)| \ 0\leq t\leq 300`

`P(0\leq X\leq 300)=-0.7408+1\\=0.2592`

Hence probability of bulb failing within 300hrs is 25.92% or 0.2592