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Suppose that the length of a phone call in minitues is an exponential rando

Suppose that the length of a phone call in minitues is an exponential random variable with parameter A=1/10.if someone arrives immediately ahead of you at a public telephone booth.find the probability that you will have to wait for.a)more than 10 min,b)between 10 and 20 min

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A)

It can be solved by Exponential Distribution 
 f(x) = λe-λx  if x> 0 
         0        if x< 0 
The cumulative distributive function F(a) of an exponential random variable is given by 
                F(a) = P(x≤a) = ∫0 λe-λx dx = 1- e-λ*a 

a) P(x>10) = 1- P(x<10) 
                = 1- (1- F(10)) 
               =  1 - (1- e-λ*10) = e-1 = 0.368 

b) P(10<X<20) =  F(20) - F(10) 
                      = (1-e-λ20) - (1-e-λ10)  = e-1  - e-2 =   .233

 

 

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