# Exponential probability density

• EV33
In summary, the problem is asking for the probability that both bulbs of a lamp, modeled by an exponential density function with a mean of 1000 hours, will fail within 1000 hours. The solution would involve finding the probability of the sum of the two bulbs' lifetimes, which would require integrating the function from 0 to 1000 for both bulbs.
EV33

## Homework Statement

a.) A lamp has two bulbs of a type with an average lifetime of 1000 hours. Assuming that we can model the probability of failure of these bulbs by an exponential density function with mean u = 1000, find the probablity that both of the lamp's bulbs fail within 1000 hours.
b.) Another lamp has just one bulb of the same type as in part (a). If one bulb burns out and is replaced by a bulb of the same type, find the probability that the two bulbs fail within a total of 1000 hours.

## Homework Equations

Exponential density function: f(t) = { 0 if t < 0
u-1e-t/u if t ≥ 0}

## The Attempt at a Solution

Plugging 1000 in for u, I got f(t) = { 0 if t < 0
1000-1e-t/1000 if t ≥ 0}

My guess is that I need to say that the function I got is the function for both light bulb #1 (I'll call it X) and light bulb #2 (I'll call it Y). Therefore, I am assuming I need to multiply these two functions X and Y together and then find when P(X+Y ≤ 1000). Therefore I would integrate the function where x goes from 0 to 1000 and y goes from 0 to 1000 - x.

Am I right in my steps cause I am really not sure if I completely understand probability density...?

Thanks.

Can anyone tell me if my steps are right?

You could have each bulb fail at 900 hours so X+Y = 1800. So X + Y < 1000 isn't what you are wanting.

So would it be X + Y < 2000 then? So my bounds would be from 0 to 2000 - x and 0 to 2000?

No. Why are you thinking about X + Y in the first place? The question isn't about a sum. You have two events, presumably independent although you didn't say so, that must both happen.

I am thinking about X + Y cause I have no idea how to approach this problem so I was just copying the other example of an exponential density function in my book.

Which part are you trying to solve, a or b?

vela said:
Which part are you trying to solve, a or b?

Good question; I'm assuming (a).

## 1. What is an exponential probability density function (PDF)?

An exponential PDF is a statistical function that describes the probability of a continuous random variable taking on a certain value within a given range. It is commonly used to model the time between events in a Poisson process, such as the time between earthquakes or the time between customer arrivals in a queue.

## 2. How is the exponential PDF different from other probability distributions?

The exponential PDF is unique in that it has a constant rate parameter, which means that the probability of an event occurring within a certain time frame is independent of when the previous event occurred. This is unlike other distributions, such as the normal distribution, which have changing probabilities based on past events.

## 3. What is the formula for the exponential PDF?

The exponential PDF is represented by the formula f(x) = λe^(-λx), where λ is the rate parameter and x is the variable of interest. This formula can be used to calculate the probability of a random variable falling within a certain range.

## 4. How is the exponential PDF used in real-world applications?

The exponential PDF is commonly used in fields such as engineering, economics, and biology to model the time between events. For example, it can be used to predict the failure rate of mechanical components or the time between customer purchases at a store.

## 5. What are the limitations of using the exponential PDF?

One limitation of the exponential PDF is that it assumes events occur independently and at a constant rate, which may not always be the case in real-world scenarios. It also does not account for rare or extreme events, as the probability of these events decreases exponentially as the time between them increases.

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