Finding the Probability Density Function

[sarujin]

Homework Statement

A dial indicator has a needle that is equally likely to come to rest at an angle between 0 and Pi. Consider the y-coordinate of the needle point (projection on the vertical axis). What is the probability density function (PDF) p(y)?

Homework Equations

I know the integral of p(y) over all space has to equal 1. The y-coordinate of the dial is of course radius*sine(theta).

The Attempt at a Solution

The first part of the question asked for the PDF for the angle, which wasn't too difficult. Knowing the integral over all space had to equal 1 and that the probability was a constant I could see that p(theta)=1/Pi . I just can't find any recipe on how to come up with the PDF for the y coordinate, in most cases it is given! I can see that it must be zero at both y=0 and y=r, which suggests to me it is probably sine or sine^2 but I cannot prove it.

Thanks a bunch.

 

[mighty2000]

Thank you for your question. I can provide some insight into how to approach this problem. First, let's define some variables:

- θ: the angle at which the needle comes to rest, between 0 and Pi
- r: the radius of the dial indicator
- y: the y-coordinate of the needle point (projection on the vertical axis)

Now, we know that the probability of the needle coming to rest at a particular angle θ is given by p(θ) = 1/Pi. This means that the probability density function for θ is a constant. But how can we find the PDF for y?

To find the PDF for y, we can use the transformation method. This method allows us to find the PDF for a new variable (in this case, y) by transforming the PDF of a known variable (in this case, θ). The transformation we will use is y = r*sin(θ). This means that for a given value of θ, the corresponding value of y can be found by multiplying r by the sine of θ.

Now, let's think about the range of values for y. We know that the needle can come to rest at any angle between 0 and Pi, so the range of values for y will be between 0 and r (since the maximum value of sin(θ) is 1). This means that the PDF for y will be zero for y < 0 and y > r, just as you mentioned.

To find the PDF for y, we can use the transformation formula:

p(y) = p(θ)/|dy/dθ|

where |dy/dθ| is the absolute value of the derivative of y with respect to θ. In this case, we have:

p(y) = p(θ)/|dr*cos(θ)|

Since p(θ) is a constant, we can replace it with 1/Pi. And since cos(θ) is also a constant, we can take it out of the absolute value sign. This gives us:

p(y) = (1/Pi)/|dr|*|cos(θ)|

Now, we know that dr is just a constant value (the radius of the dial indicator), so we can take it out of the absolute value sign as well. This gives us:

p(y) = (1/Pi)*|cos(θ)|