Properties of Exponential Distributions

An exponential distribution is:

• A highly used ‘continuous’ distribution
• Often used to model the time elapsed between events (we call this [latex]x[/latex]).
• It is ‘memoryless’ (see more in the ‘MEMORYLESS’ section below)
• We call the average lambda ([latex]\lambda[/latex]). It measures the average number of events per time unit.
• It is right skewed (mean > median), and [latex]\lambda[/latex] and [latex]x[/latex] can never be negative.

Calculating Probabilities

We can calculate the probabilities using formulas or Excel. The formulas are:

• [latex]P(\text{at most or less than}) = P(X \le x) = 1 - e^{-\lambda x}[/latex]
• [latex]P(\text{at least or more than}) = P(X \ge x) = e^{-\lambda x}[/latex]

The Excel calculations are:

• [latex]P(\text{at most or less than}) =\text{EXPON.DIST}(x, \lambda, \text{TRUE})[/latex]
• [latex]P(\text{at least or more than}) =1-\text{EXPON.DIST}(x, \lambda, \text{TRUE})[/latex]

Memoryless Property

• Memoryless means that it does not matter how much time has elapsed previously.
• When calculating the odds of a certain amount of time elapsing going forward,
• It’s although the clock resets to zero.
• See Wikipedia‘s explanation of memoryless:
  

  It describes situations where the time you’ve already waited for an event doesn’t affect how much longer you’ll have to wait

Statistical Properties

The following metrics apply to exponential distributions:

• [latex]x = \text{time between events}[/latex]
• [latex]\lambda = \text{lambda}=\text{number of events per time unit}[/latex]
• The mean is: [latex]\mu = \frac{1}{\lambda}[/latex]
• The standard deviation is also: [latex]\frac{1}{\lambda}[/latex]
• The variance is: [latex]\sigma^2 = \frac{1}{\lambda ^2}[/latex]
• The distribution is right skewed and the skewness = 2.

Video Explaining Exponential Distributions

Key Takeaways (EXERCISE)

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