Probability

Question

Suppose that the length of the phone calls in minutes is an exponential random variable with parameter $\lambda = 1/10$.

If someone arrives immediately ahead of you at a public telephone booth, find the probability of that you will have to wait

a) more than $10$ minutes (ans $0.368$)
b) between $10$ and $20$ minutes (ans $0.233$)

Answer 1

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^a  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

Comments

$F(x)$ is generally used for CDF.

why not putting P(X=1) + P(X=2) + ..... + P(X=9) instead of F(1) + F(2) ..........+F(9)

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right i did mistake in hurry.