Probability equation in case of an exponential distribution

In summary: Therefore, the question is not about the particle's specific lifetime, but about the probability of a certain event (the particle's lifetime being longer than a certain time) occurring. And in this case, the memoryless property does not hold, as the probability of the event changes depending on whether the particle has already existed for a certain amount of time. This is why we cannot simply write ##P(X>5|X>1) = P(X>5)## in the case of an exponential distribution. In summary, the memoryless property of the exponential distribution means that the past has no influence on future probabilities, but it does not mean
  • #1
user366312
Gold Member
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Why is ##P(X>5|X>1) = P(X>4)## in case of an exponential distribution?

Can anyone kindly explain it to me?
 
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  • #2
Just write out the densities probabilities using density of exponential distribution and it will become clear.
 
  • #3
Math_QED said:
Just write out the densities probabilities using density of exponential distribution and it will become clear.

I couldn't understand.
 
  • #4
user366312 said:
I couldn't understand.

Please define exponential distribution.
 
  • #5
Math_QED said:
Please define exponential distribution.

The density function for an exponential distribution is ##f(x; \lambda) = \lambda e ^{-\lambda x}## in case of ##x\ge0##.

The plot would be something like the following:

241846
 
  • #6
Okay, good. Do you know that ##P(X \leq t) = \int_{0}^t \lambda e^{-\lambda x} dx## for ##t > 0##? If so, why don't you just calculate all probabilities involved and see that they are equal?
 
  • #7
Math_QED said:
Okay, good. Do you know that ##P(X \leq t) = \int_{0}^t \lambda e^{-\lambda x} dx## for ##t > 0##? If so, why don't you just calculate all probabilities involved and see that they are equal?

1. Yes, I know.
2. I want need to understand it either graphically or by proof in order to solve understand the following problem as I already have the solution with me:

241847
 
  • #8
user366312 said:
1. Yes, I know.
2. I want need to understand it either graphically or by proof in order to solve understand the following problem as I already have the solution with me:

View attachment 241847

I don't understand your lack of understanding. You have formulas for P(X>5|X>1)---just the ordinary conditional probability formula. So, if you know how to compute the numerator P(X > 5 & X > 1) you are done.
 
  • #9
Ray Vickson said:
I don't understand your lack of understanding. You have formulas for P(X>5|X>1)---just the ordinary conditional probability formula. So, if you know how to compute the numerator P(X > 5 & X > 1) you are done.

##P(X>5|X>1) = \frac{P(X>5, X>1)}{P(X>1)}##

Is this the correct formula?
 
  • #10
Do not forget exponential distribution has a memoryless property. When you set a condition, you do not have to consider the cases before it.
 
  • #11
mertcan said:
Do not forget exponential distribution has a memoryless property. When you set a condition, you do not have to consider the cases before it.

So, can I write ##P(X>5|X>1) = P(X>5)## ?
 
  • #12
user366312 said:
##P(X>5|X>1) = \frac{P(X>5, X>1)}{P(X>1)}##

Is this the correct formula?

What do YOU think? What is stopping you from carrying on, to complete the task?
 
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  • #13
Ray Vickson said:
What do YOU think? What is stopping you from carrying on, to complete the task?

As I have already said, I have calculations with me. I am trying to understand the core idea.

If exponential distribution is memoryless (i.e. the past has no bearing on its future behavior), why can't I write ##P(X>5|X>1) = P(X>5)##?
 
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  • #16
user366312 said:
So, can I write ##P(X>5|X>1) = P(X>5)## ?

No, obviously not: the left-hand side evaluates to ##P(X > 4),## and that is different from ##P(X > 5)##.

Look at it intuitively: once you have reached time ##1## you have 4 units left to go to reach time ##5##. The memoryless property says that the probability of having to wait at least 4 is exactly the same as for a brand-new random variable X that that has just been born at time 0----in other words, you have the same P(X > 4) whether you have just started or have already waited 1 unit (or 50 units or ... ).

The easiest way to write the memoryless property is to say that
$$P(X > t+s | X > t) = P(X > s)$$ So, if ##t=1## and ##t+s=5## we have ##s=4.##
 
Last edited:
  • #17
user366312 said:
As I have already said, I have calculations with me. I am trying to understand the core idea.

If exponential distribution is memoryless (i.e. the past has no bearing on its future behavior), why can't I write ##P(X>5|X>1) = P(X>5)##?

I'll describe what I think of this. Probability theory does not generally deal with the calculation of probabilities, but with an extraordinary task. That's it
  1. examining the laws of random mass phenomena,
  2. under precisely defined experimental conditions.
.................................
Suppose that the expected lifetime of a radioactive particle is ##1/\lambda##. In this case (I hope I remember it well) the distribution of its lifetime is exponential. The lack-of-memory property means that the particle is not aging, so if the particle has lived for a hundred years, it will have the same probability of surviving another hundred years as if it had just emerged in an atomic process.

The question, however, was whether the probability that the particle lifetime would be longer than five years provided that it had already lived for more than one year would be the same as the probability that the initially generated particle would last for more than five years. He forgets his past - that he already existed for a year - so it might seem right. On the other hand, the statement is false.

Calculate the probabilities.
$$ P(X>5) = e^{-5\lambda}, $$
$$ P(X>5 | X>1) = \frac {P(X>5~ and~ X>1)} { P(X>1)} = \frac {P(X>5)}{ P(X>1)} = e^{-5\lambda}/e^{-\lambda} = e^{-4\lambda}. $$
The two quantities are different.

.................................
However, as I wrote above, probabilities can only be discussed under predetermined experimental conditions.
If the experimental conditions are fixed so that only the random events in which the particle has lived for more than one year are tested, the same probability is that it will live for more than five years, provided it lived for more than one year, as the probability that the newly formed particle will live longer than five years. However, if we examine not only the particles that live longer than one year, but all the new particles, then these two quantities are different. The above calculation was for this.

The lack-of-memory property doesn't mean we forget what the experimental conditions are, which we are investigating.
 
  • #18
Periwinkle said:
I'll describe what I think of this. Probability theory does not generally deal with the calculation of probabilities, but with an extraordinary task. That's it
  1. examining the laws of random mass phenomena,
  2. under precisely defined experimental conditions.
.................................
Suppose that the expected lifetime of a radioactive particle is ##1/\lambda##. In this case (I hope I remember it well) the distribution of its lifetime is exponential. The lack-of-memory property means that the particle is not aging, so if the particle has lived for a hundred years, it will have the same probability of surviving another hundred years as if it had just emerged in an atomic process.

The question, however, was whether the probability that the particle lifetime would be longer than five years provided that it had already lived for more than one year would be the same as the probability that the initially generated particle would last for more than five years. He forgets his past - that he already existed for a year - so it might seem right. On the other hand, the statement is false.

Calculate the probabilities.
$$ P(X>5) = e^{-5\lambda}, $$
$$ P(X>5 | X>1) = \frac {P(X>5~ and~ X>1)} { P(X>1)} = \frac {P(X>5)}{ P(X>1)} = e^{-5\lambda}/e^{-\lambda} = e^{-4\lambda}. $$
The two quantities are different.

.................................
However, as I wrote above, probabilities can only be discussed under predetermined experimental conditions.
If the experimental conditions are fixed so that only the random events in which the particle has lived for more than one year are tested, the same probability is that it will live for more than five years, provided it lived for more than one year, as the probability that the newly formed particle will live longer than five years. However, if we examine not only the particles that live longer than one year, but all the new particles, then these two quantities are different. The above calculation was for this.

The lack-of-memory property doesn't mean we forget what the experimental conditions are, which we are investigating.

@Periwinkle Thanks for the answer! That concludes my query.
 

What is an exponential distribution?

An exponential distribution is a probability distribution that describes the time between events in a Poisson process. It is often used to model the time between arrivals of customers in a queue or the time between failures of a machine.

What is the probability equation for an exponential distribution?

The probability density function (PDF) for an exponential distribution is given by the equation f(x) = λe^(-λx), where λ is the rate parameter and x is the time between events. The cumulative distribution function (CDF) is given by F(x) = 1 - e^(-λx).

How is the rate parameter related to the shape of the exponential distribution?

The rate parameter, λ, determines the shape and scale of the distribution. A larger value of λ results in a steeper decline of the PDF and a shorter average time between events. A smaller value of λ results in a flatter decline of the PDF and a longer average time between events.

What is the mean and standard deviation of an exponential distribution?

The mean of an exponential distribution is equal to 1/λ, and the standard deviation is also equal to 1/λ. This means that the average time between events is equal to the inverse of the rate parameter, and the spread of the distribution is also determined by the rate parameter.

How can the exponential distribution be used in real-world applications?

The exponential distribution is commonly used in fields such as queuing theory, reliability analysis, and financial modeling. It can be used to model the time between events in various systems, such as customer arrivals, machine failures, or stock price changes. It can also be used to calculate probabilities of events occurring within a certain time frame.

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