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. Use this model to find the probability that a bulb (i) fails within the first 200 hours, (ii) burns for more than 800 hours.

Answer 1

Answer:

(i) 0.1812

(ii) 0.4493

Step-by-step explanation:

if the time of failure of a lightbulb is described by an exponential density function, the function of density f(x) is given by:

$f(x)=\frac{1}{1000} *e^{\frac{-x}{1000}}$ for x≥0

Where x is the time failure

Additionally, the distribution function F(x), that give us the probability that the light bulb fail in a time less or equal than x is given by:

$F(x)=1-e^{\frac{-x}{1000} }$  for x≥0

So, the probability that a bulb fails within the first 200 hours is the probability that the bulb fail in a time less or equal to 200. That is calculate as:

$F(200)=1-e^{\frac{-200}{1000} }=0.1812$

Then, the probability that a bulb burns for more than 800 hours is the complement of the probability that the bulb fail in a time less or equal to 800. That is calculate as:

$1-F(800)=1 - (1-e^{\frac{-800}{1000} })=0.4493$