# Continuous Random Variables

## The Exponential Distribution (Optional)

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long-distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.
Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.
Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.

### Memorylessness of the Exponential Distribution

In  recall that the amount of time between customers is exponentially distributed with a mean of two minutes (X ~ Exp(0.5)). Suppose that five minutes have elapsed since the last customer arrived. Since an unusually long amount of time has now elapsed, it would seem to be more likely for a customer to arrive within the next minute. With the exponential distribution, this is not the case—the additional time spent waiting for the next customer does not depend on how much time has already elapsed since the last customer. This is referred to as the memoryless property. Specifically, the memoryless property says the following
For example, if five minutes have elapsed since the last customer arrived, then the probability that more than one minute will elapse before the next customer arrives is computed by using r = 5 and t = 1 in the foregoing equation.
This is the same probability as that of waiting more than one minute for a customer to arrive after the previous arrival.
The exponential distribution is often used to model the longevity of an electrical or a mechanical device. In , the lifetime of a certain computer part has the exponential distribution with a mean of ten years (X ~ Exp(0.1)). The memoryless property says that knowledge of what has occurred in the past has no effect on future probabilities. In this case it means that an old part is not any more likely to break down at any particular time than a brand new part. In other words, the part stays as good as new until it suddenly breaks. For example, if the part has already lasted ten years, then the probability that it lasts another seven years is P(X > 17|X > 10) = P(X > 7) = 0.4966.

### Relationship Between the Poisson and the Exponential Distribution

There is an interesting relationship between the exponential distribution and the Poisson distribution. Suppose that the time that elapses between two successive events follows the exponential distribution with a mean of μ units of time. Also assume that these times are independent, meaning that the time between events is not affected by the times between previous events. If these assumptions hold, then the number of events per unit time follows a Poisson distribution with mean λ = 1/μ. Recall from the chapter on Discrete Random Variables that if X has the Poisson distribution with mean λ, then 
. Conversely, if the number of events per unit time follows a Poisson distribution, then the amount of time between events follows the exponential distribution. (k! = k*(k–1*)(k–2)*(k–3)…3*2*1)

### Chapter Review

If X has an exponential distribution with mean μ, then the decay parameter is m = 
, and we write X ~ Exp(m) where x ≥ 0 and m > 0 . The probability density function of X is f(x) = me (or equivalently 
 . The cumulative distribution function of X is P(X ≤ x) = 1 – e–.
The exponential distribution has the memoryless property, which says that future probabilities do not depend on any past information. Mathematically, it says that P(X > x + k|X > x) = P(X > k).
If T represents the waiting time between events, and if T ~ Exp(λ), then the number of events X per unit time follows the Poisson distribution with mean λ. The probability density function of X is 
. This may be computed using a TI-83, 83+, 84, 84+ calculator with the command poissonpdf(λ, k). The cumulative distribution function P(X ≤ k) may be computed using the TI-83, 83+,84, 84+ calculator with the command poissoncdf(λ, k).

### Formula Review

Exponential: X ~ Exp(m) where m = the decay parameter
1. pdf: f(x) = me(– where x ≥ 0 and m > 0
2. cdf: P(X ≤ x) = 1 – e(–
3. mean µ = 


4. standard deviation σ = µ
5. percentile k: k = 
 

6. Additionally


7. Memoryless property: P(X > x + k|X > x) = P (X > k)
8. Poisson probability: 
 with mean λ
9. k! = k*(k−1)*(k−2)*(k−3)*…3*2*1

### References

Baseball-Reference.com. (2013). No-hitter. Retrieved from http://www.baseball-reference.com/bullpen/No-hitter
U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/
World Earthquakes. (2013). Earthquake data for Papua New Guinea. Retrieved from http://www.world-earthquakes.com/
Zhou, Rick. (2013). Exponential distribution lecture slides. Retrieved from www.public.iastate.edu/~riczw/stat330s11/lecture/lec13.pdf
Use the following information to answer the next 10 exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)
Use the following information to answer the next eight exercises. A distribution is given as X ~ Exp(0.75).
Use the following information to answer the next eight exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.

### Homework

Use the following information to answer the next three exercises. The average lifetime of a certain new cell phone is three years. The manufacturer will replace any cell phone failing within two years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution.