Exponential Distribution

Definition:

• Continuous random variable
• Memoryless Variable
  • inherits $P(X>s+t)=P(X>s).P(X>t)$
• Describes the time between events that occur randomly and independently at a constant average rate.
• The number of occurrence in a timeframe is $\lambda$
• It is often used to model waiting times, such as the time it takes for a customer to be served at a checkout counter or the time it takes for a website to load.

Probability density function:

• For some $\lambda >0$ by $f(x)=\{\begin{array}{ll}\lambda e^{-\lambda x}& \text{for }x\ge 0\\ 0& \text{for }x<0\end{array}$

• $X\sim Exp(1/8)$

Expected value $E[X]=\frac{1}{\lambda }$

Variance $Var(X)=\frac{1}{{\lambda }^2}$

Cumulative distribution function:

• $F(a)=P(X\le a)=1-e^{-\lambda a},a\ge 0$

2 exponential variables:

• Probability that one exponential random variable is smaller than another:
• $X_1$ and $X_2$ are independent Exponential random variable with means $1/{\lambda }_1$ and $1/{\lambda }_2,P(X_1<X_2)=\frac{{\lambda }_1}{{\lambda }_1+{\lambda }_2}$

Hazard rate function

• Suppose now that the lifetime distribution is exponential
• Then, by the memory-less property, it follows that the distribution of remaining life for a $t-$year-old item is the same as that for a new item.
• Hence, λ(t) should be constant. In fact, this checks out, since $\lambda (t)=\frac{f(t)}{\bar{F}(t)}=\frac{\lambda e^{-\lambda t}}{e^{-\lambda t}}=\lambda$
• Thus, the failure rate function for the exponential distribution is constant.