Exponential Distribution and Properties

– Stochastic Process course notes

1. Definition

  • Probability function: f(x) = lambda e^{-lambda x}, x geq 0
  • Cumulative Distribution Function (CDF): F(x) = 1-e^{-lambda x}, x geq 0
  • Complement of the CDF (CCDF): F^c(x) = e^{-lambda x}, x geq 0.

2. Memoryless Property

Def`1: A random variable X has the memoryless property if Pr{X>t+s| X>s} = Pr{X>t}

Def`2A random variable X has the memoryless property if Pr{X>t+s} = Pr{X>t} Pr{X>s}

The exponential distribution is the only distribution that has the memoryless property (Satisfy definition 2)

3. Useful Properties: First occurrence among events

Assume X_1, X_2, cdots, X_n are exponential variable with rate lambda_1, lambda_2, cdots, lambda_n.
Then what is the probability that X_1 < X_2.

More generally

Pr{X_i = min[X_1,cdots, X_n]} = frac{lambda_i}{lambda_1+lambda_2+cdots +lambda_n}

4. Distribution of time of first event

This is the CDF of an exponential RV with rate (lambda_1 + lambda_2), therefore
min(X_1, X_2) ~ exp(lambda_1 + lambda_2)

5. Distribution of time of last event (maximum)

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