Answer:
[tex]0.2592 \ or \ 25.92\%[/tex]
Explanation:
The exponential density function is given as
[tex]f(t)=\left \{ {{0} \atop {ce^{ct}}} \right\\0,t<0\\ce^{ct},t\geq 0[/tex]
[tex]\mu=\frac{1}{c}\\c=\frac{1}{\mu}\\\\=\frac{1}{1000}=0.001\\\\f(t)=0.001e^{-0.001t}[/tex]
To find probability that bulb fails with the first 300hrs, we integrate from o to 300:
[tex]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[/tex]
[tex]P(0\leq X\leq 300)=-0.7408+1\\=0.2592[/tex]
Hence probability of bulb failing within 300hrs is 25.92% or 0.2592
