The lifespan of a car battery averages six years. Suppose the batterylifespan follows an exponential distribution.(a) Find the probability that a randomly selected car battery will lastmore than four years.(b) Find the variance and the 95th percentile of the battery lifespan.(c) Suppose a three-year-old battery is still going strong. (i) Find theprobability the battery will last an additional five years. (ii) Howmuch longer is this battery expected to last

Step-by-step explanation:

Let X denote the life span of a car battery and it follows and exponential distribution with average of 6 years.

Thus , the parameter of the exponential distribution is calculated as,

μ = 6

a) The required probability is

= 0.513

Hence, the probability that a randomly selected car battery will last more than four years is 0.513

b) The variance of the battery span is calculated as

The 95% percentile (α = 5%) of the battery span is calculated

c)  

Let denote the remaining life time of a car battery

i)the probability the battery will last an additional five years is calculated below

ii) The average time that the battery is expected to last is calculated

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