Probability density function
A continuous random variable whose probability density function is given for some k > 0 by
\[F(x)=ke ^{-kx} \] if x > 0 or equal to 0
and f(x)=0 if x< 0 is said to be exponential random variable with parameter k .
Cumulative distribution
Cumulative distribution function f (a,b) is given by P( \[a\leq x\leq b\])=\[\int_{a}^{b} ke^{-kx}\]=\[(1-e^{-kb})-(1-e^{-ka})\]
For exponential distribution Mean or expectation E[X]= \[\frac{1}{k}\]
Variance V(x)=\[\frac{1}{k^2}\]
For Example . Suppose the length of phone call is minutes is exponential distribution with k=0.1. if someone arrives immediately ahead of you at public telephoe booth, find the probability that you will have to wait (a) more than 10 mins (b) between 10 to 20 mins.
solution.
(a) P{X >10} = 1- p(x <10)= 1- f(10)=1- (1-e^(-10k)=0.368 here a=0 k=0.1 b=10
(b) here a=10 b=20 k=0.1 hence output=0.233
