Exponential Distribution

Introduction

TBA

Density function

The density function for the Exponential distribution is $$f(x | \lambda ) = \lambda e^{-\lambda x}$$

The density function for the Poisson distribution depends on the rate parameter $\lambda =$ $\, 1.5 \,$.

Specific to the plot above, the density function is $$f(x | \lambda = 1.5) = 1.5 e^ { -1.5 x }$$

Examples

1. In a mechanical engineering workshop, ball bearings are used in various machinery. The lifetime, $T$ (in hours), of a specific type of ball bearing is known to follow an Exponential distribution with a mean lifetime of 5,000 hours.
   1. What is the probability that one randomly selected ball bearing will last more than 6,000 hours?
   2. What is the probability that one randomly selected ball bearing will last less than 2,500 hours?
2. A major manufacturer of capacitors claims that their latest model, the UltraCap 3000, has an average lifetime of 5,000 hours before it fails. This lifetime is exponentially distributed. Capacitors that fail prematurely can cause significant issues in electrical circuits, so understanding their reliability is crucial.
   1. What is the probability that the UltraCap 3000 will last more than 7,000 hours before failing?
   2. If an electrical engineer installs three UltraCap 3000 capacitors in parallel in a circuit (assuming they operate independently), what is the probability that at least one will fail within 3,000 hours?
   3. On average, how long will it take for 10% of these capacitors to fail?
3. Civil engineers are tasked with ensuring that bridges are safe and functional for the duration of their expected lifetimes. An old city bridge is being inspected, and it is known from past data that the time (in years) until a significant structural defect appears on this type of bridge follows an Exponential distribution. Suppose the average time until a significant defect appears is 20 years.
   1. What is the probability that a significant defect will appear in the first 10 years?
   2. How long until there is a 50% chance of a significant defect appearing?
4. Suppose a particular radioactive isotope has a mean lifetime of $\tau$ seconds. The time $T$ until the isotope decays follows an Exponential distribution with parameter $\lambda = 1 / \tau$.
   1. What is the probability that the isotope decays in less than $\tau / 2$ seconds?
   2. If a physicist has a sample containing 1000 such isotopes, approximately how many isotopes would she expect to decay within $\tau / 2$ seconds?
5. Imagine you are a math educator and you've just introduced the concept of the Exponential distribution to your students. To emphasize real-world applications, you provide the following example: The time between a student asking a question in class follows an Exponential distribution with a mean of 10 minutes.
   1. What is the probability that the time between two questions is less than 5 minutes?
   2. What is the probability that the time between two questions is between 5 and 15 minutes?

Calculator

$X \sim\;$Exponential( $\lambda =$ $\, 1.5 \,$). $\mathbb{P}[X$ $\le$ $\, 1 \,$ $] = 0.78$.

References

Exponential distribution. Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. Accessed 17/10/2023.