probability distribution of a function

[Richard Baker]

I have a function y (x) defined over a range from x₀ to x₁ 

I want to know the probability that y will equal some number within a given interval if I choose a random value for x. 

 

[exchemist]

6 minutes ago, Richard Baker said:

I have a function y (x) defined over a range from x₀ to x₁ 

I want to know the probability that y will equal some number within a given interval if I choose a random value for x. 

 

But surely for any given value of x, y has a defined value, doesn't it?  

[Genady]

If you can solve inequality y(x)<b, then the range of x in this solution over x₁-x₀ gives you probability P(y<b). The probability P(a<y<b)=P(y<b)-P(y<a). 

[studiot]

I think the question is ill posed.

 

P(anything) is positive or zero.

But the conditions given could include y(x) < 0 for all or some x.

[Genady]

1 minute ago, studiot said:

I think the question is ill posed.

 

P(anything) is positive or zero.

But the conditions given could include y(x) < 0 for all or some x.

I understand that they ask about, in this case, P(y(x) < 0), which is in [0,1].

Example: y(x) = x², on [0, 10]. What is probability of y to be within [36, 64]?

For x² < b, x < sqrt(b).

So, P(36<y<64) = (8 - 6)/10 = .2

[studiot]

20 minutes ago, Genady said:

I understand that they ask about, in this case, P(y(x) < 0), which is in [0,1].

Example: y(x) = x², on [0, 10]. What is probability of y to be within [36, 64]?

For x² < b, x < sqrt(b).

So, P(36<y<64) = (8 - 6)/10 = .2

How is that what Richard asked ?

ie how is the the probability  P(y=g), which is the given question ?

 

Edit Yes I see what you are saying now.

+1

Edited August 18, 2023 by studiot

[Genady]

1 minute ago, studiot said:

How is that what Richard asked ?

ie how is the the probability  P(y=g), which is the given question ?

He asked about

 

44 minutes ago, Richard Baker said:

some number within a given interval

(my emphasis)

Edit: x-posted 

[Leojames26]

You’ve got a function y(x)y(x)y(x) over a range from x0x_0x0 to x1x_1x1, and you want to find the chance that yyy will be a certain value or within a certain interval if you randomly pick a value of xxx.

To figure that out, you would:

1. Look at the range of yyy values you care about.
2. Find out which xxx values give you those yyy values.
3. See how big that xxx range is.
4. Then, just divide that range by the total xxx range (from x0x_0x0 to x1x_1x1).

That’s your probability. It’s basically asking, “Out of all possible xxx values, how many give me a yyy that’s in the interval I care about?”

If you’ve got more specifics about the function or the interval, it could make this clearer, but that’s the general approach!