The lifetime in hours of a lightbulb is an exponentially dis

The lifetime (in hours) of a lightbulb is an exponentially distributed random variable with parameter = 0.01 (units of hours¹).

(a) What is the probability that the light bulb is still burning one week after it is installed?

(b) Assume that the bulb was installed at noon today and assume that at 3:00pm tomorrow you notice that the bulb is still working. What is the chance that the bulb will burn out at some time between 4:30pm and 6:00pm tomorrow?

**Please provide all calculations and exlaination to answers!

Solution

a)

          
The right tailed area in an exponential distribution is          
          
Area = e^(-lambda*t)          
          
As          
          
t = critical value = 24 hrs*7 =     168 hours
          
          
Then          
          
Area =    0.186373976   [ANSWER]

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B)

Note that the exponential distribution is memoryless, so that means we can treat time 3:00pm to be t = 0, and t1 = 4:30pm = 1.5 hrs, t2 = 6pm = 3 hrs.

The area between two values in an exponential distribution is          
          
Area = e^(-lambda*t2) - e^(-lambda*t1)          
          
As          
          
x1 = lower bound =   1.5      
x2 = upper bound =    3      
          
Then          
          
Area =    0.014666406   [ANSWER]