Exponential random variable

Exponential random variable

The exponential random variable is the continuous analogue of the geometric distribution. This random variable has the key property of being memoryless meaning that the exponential random variable $T$ has the following property: $$ P(T>t+s)=P(T>t)P(T>s) $$ The exponential random variable is the solution of the Kolmogorov equation for the two-state jump rate matrix: $$ \mathbf{Q} = \left( \begin{matrix} -\lambda & \lambda \\ 0 & 0 \end{matrix} \right) $$ The probability distribution function for this random variable is: $$ F_T(t)= P(T\le t) = 1 - e^{-\lambda t} $$ while the probability density is: $$ f_T(t) = \lambda e^{-\lambda t} $$ The expectation and variance of this random variable are $\mathbb{E}(T)=\frac{1}{\lambda}$ and $\textrm{var}(T) = \frac{1}{\lambda^2}$

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