Exponential Distribution

left=-1; right=5;
bottom=-1;top=2;
---
y = e^{-x} \{ x \ge 0 \}
y = 0 \{ x < 0 \}

Support and CDF

(support and c.d.f.)

left=-1; right=5;
bottom=-1;top=2;
---
y = 1 - e^{-x} \{ x \ge 0 \}
y = 0 \{ x < 0 \}

Alternate notation: in the STAT 231 notes, and

Expectation and Variance

(expected value and variance)

Memorylessness

1. Waiting time interpretation
   • Detector for radioactive decay
     • Waiting for detector to go off has same probability if you came in later

Relationships to Other Distributions

1. Bernoulli Distribution
   • Sequence of 1 and 0
   • "Waiting time" for first success is the Geometric Distribution
2. Poisson Distribution
   • Poisson process: continuous version of Bernoulli
   • "Waiting time" for first success is the exponential distribution

proof
Let CDF of be . Looking at :

• : probability you had to wait more than for first success
• : this is Poisson, which measures the number of successes in some interval, so

todo what??!??

Examples

Note: "percentile" refers to an actual value.

[!example]
City has busses that appear every 10 minutes, but you arrive at bus stops at random times (uniform).

1. What is the expected waiting time?
2. Find

solution
Let be the time spend waiting in minutes.

1. Should wait on average minutes

[!example]
City has busses that appear , but you arrive at bus stops at random times (uniform).

1. What is the expected waiting time?
2. Find

solution
Let be the waiting time in minutes

2. ???