Probability over time interval?

[Destroxia]

Homework Statement

This is a problem I have posed for myself.

Suppose I have found the probability of something occurring over an angle, so it is the probability of landing at a particular angle over an interval of 2 angles.

How would I now find the probability of landing there within a specific amount of time? Say I wanted to find the probability of landing on that angle over a certain time interval. So it is sort of like, an interval within a time interval.

The Attempt at a Solution

I have already found the probability of it landing somewhere in the interval using a probability density function, now I want to add a time variable into the landing of it. Do I need to define some sort of constant which says what rate it is moving from point a to b on the interval?

 

[andrewkirk]

Is there only one possible object - call it O - that could land? If so, then the probability of it landing with an angle in the range [a1,a2] during the time interval [t1,t2] is

$$P(A\cap B)=P(A|B)P(B)$$

where ##B## is the event that O lands in the time interval [t1,t2] and A is the event that O lands with angle in interval [a1,a2].
If the angle of landing is independent of the time of landing then this simplifies to ##P(A)P(B)##. Otherwise, you need an expression for the dependence between the two events before you can go any further.

If there is more than one possible object that can land then:

1. You need to use an arrival distribution like the Poisson, to generate the number of landings in the interval [t1,t2], given a landing frequency ##\lambda## per unit time; and
2. You need to refine your question, as you can no longer refer to 'it' landing. For instance it could be
a. What's the probability that at least one object lands at an angle in [a1,a2] in time interval t1,t2]; or
b. What's the probability that at least one object lands in time interval t1,t2] and the first such object lands at an angle in [a1,a2]; or
c. What's the probability that at least one object lands in time interval t1,t2] and the all such objects land at an angle in [a1,a2]; or
d. a lot of other possible questions.

 

[Destroxia]

Is there only one possible object - call it O - that could land? If so, then the probability of it landing with an angle in the range [a1,a2] during the time interval [t1,t2] is

$$P(A\cap B)=P(A|B)P(B)$$

where ##B## is the event that O lands in the time interval [t1,t2] and A is the event that O lands with angle in interval [a1,a2].
If the angle of landing is independent of the time of landing then this simplifies to ##P(A)P(B)##. Otherwise, you need an expression for the dependence between the two events before you can go any further.

If there is more than one possible object that can land then:

1. You need to use an arrival distribution like the Poisson, to generate the number of landings in the interval [t1,t2], given a landing frequency ##\lambda## per unit time; and
2. You need to refine your question, as you can no longer refer to 'it' landing. For instance it could be
a. What's the probability that at least one object lands at an angle in [a1,a2] in time interval t1,t2]; or
b. What's the probability that at least one object lands in time interval t1,t2] and the first such object lands at an angle in [a1,a2]; or
c. What's the probability that at least one object lands in time interval t1,t2] and the all such objects land at an angle in [a1,a2]; or
d. a lot of other possible questions.

So the equation essentially says the probability of them both occurring ( ##P(A\cap B)## ) is equal to the probability of A happening after getting the probability of B ( ##P(A|B)## ) times the probability of B occurring ( ##P(B)## ).

What is the reasoning behind this? (I'm fairly new to probability, still trying to understand some notation and thought process, but I think I'm getting the general gist)

So, what happens if my probability density function in general is impossible to define because the limit as it goes to infinity in both directions is undefined? Would that mean I have an non-continuous probability density, since my function doesn't particularly end (AKA the ##\int_{-\infty}^{\infty}p(x)dx \neq 1##)

What I mean by this is that my "object" could choose to never come to an arrival potentially anywhere along the function, or it could choose to come to rest on the function, essentially at any time it wants. So I want to be able to solve for the probability of it deciding to come to rest over a specified time interval. Say like, over an hour, what is the probability this object will decide to land anywhere on the function, given a specified angle between two points. This angle can keep repeating itself essentially.

 

[andrewkirk]

Failure rate in the continuous sense

So the equation essentially says the probability of them both occurring ( ##P(A\cap B)## ) is equal to the probability of A happening after getting the probability of B ( ##P(A|B)## ) times the probability of B occurring ( ##P(B)## ).

What is the reasoning behind this? (I'm fairly new to probability, still trying to understand some notation and thought process, but I think I'm getting the general gist)

It's how ##P(A|B)## is defined. That's called the conditional probability of A, given B, and is defined as ##P(A|B)\equiv\frac{P(A\cap B)}{P(B)}##.

So, what happens if my probability density function in general is impossible to define because the limit as it goes to infinity in both directions is undefined? Would that mean I have an non-continuous probability density, since my function doesn't particularly end (AKA the ##\int_{-\infty}^{\infty}p(x)dx \neq 1##)

If the integral is not 1 then it's not a probability. Part of the definition of a probability is that its integral must be 1.

What I mean by this is that my "object" could choose to never come to an arrival potentially anywhere along the function, or it could choose to come to rest on the function, essentially at any time it wants. So I want to be able to solve for the probability of it deciding to come to rest over a specified time interval. Say like, over an hour, what is the probability this object will decide to land anywhere on the function, given a specified angle between two points. This angle can keep repeating itself essentially.

It sounds like what you are trying to model is 'Mean Time Before Failure', where here 'failure' means landing. Here's the wiki article on it. The exponential failure distribution is a simple model that is widely used, and may suit your purpose. That's described under the heading 'Failure rate in the continuous sense'. It gives the cdf, and the pdf is the derivative of that.

 

[Destroxia]

Thank you, that sounds exactly like what I am trying to model, the exponential failure distribution that is. The object in the case of my scenario is more of a decision, but I think this might work nicely.