The Average Life of a Light Bulb Follows an Exponential Distribution with a Mean of 500 Hours

The average life of a light bulb follows an exponential distribution with a mean of 500 hours.

Find:

a) the probability that a bulb will last more than 1,000 hours

b) the probability that a bulb will last less than 100 hours

c) the probability that a bulb will last exactly 500 hours

Solution: Exponential distribution: the distribution is given by

F(x) = e -x for x>0

Mean = 1/ Standard deviation=1/

Given mean =500

Mean =1/ =500

=1/500 = 0.002

x~ ExP()

1> the probability that a bulb will last more than 1,000 hours

P(x>1000) =

= 0.002 (e -0.002x ) / 0.002

= e – 0.002(1000) – e -0.002()

P(x>1000) = e -2

2> the probability that a bulb will last less than 100 hours

P(x<100) = 0.002 e -0.002x dx

= 0.002[e -0.002 x / 0.002

= e -0.002(0) – e -0.002(100)

P(x<100) = 1 – e -0.2

3> the probability that a bulb will last exactly 500 hours

P(x=500) = 0.002 e -0.002x dx

e -0.002 x = 0

P(x=500) = 0