#### Transcript Exponential Distribution

```Exponential Distribution
• The RV T has an exponential distribution with rate
l (for l > 0) if T has the probability density
f t   l elt
t  0
• The mean and standard deviation of T are
E T  
1
l
SD T  
1
l
Exponential Distribution
• The exponential distribution is used to model
waiting times for the occurrence of some event
(death, failure, mutation, radioactive decay, etc.).
• The continuous analog of the geometric
distribution.
• Models the successive inter-arrival times of a
Poisson process in time.
• Suppose a particular kind of radioactive
atom has a half-life of 2 years. Find
– The probability that an atom of this type
survives at least 5 years.
– The time at which the expected number of
atoms is 10% of the original amount.
Gamma Distribution
• If Tr is the time of the rth arrival after t = 0 in a
Poisson process with rate l, then Tr has the
gamma (r, l) distribution with probability density
f t   le
lt
lt 
r 1!
r 1
• The mean and standard deviation of Tr are
E T  
r
l
SDT  
r
l
```