Undergrad What Does The Probability Density Function Tell You?

[tomtomtom1]

Hello All

I was wondering if someone could help explain what the probability density function tells you.

I am trying to learn about surveying and the PDF keeps cropping up and I do not fully understand it.

For example I have:-

• measured a single angle 15 times
• calculated my Standard Deviation to be 0.61
• the Mean to be 53.61
• Used the Probability Density Function to calculate Density for each of my measurements.

The results are as follows:-

Measurements

1) 52.7
2) 53.0
3) 53.0
4) 53.4
5) 53.4
6) 53.4
7) 53.4
8) 53.4
9) 53.4
10) 53.7
11) 53.7
12) 54.0
13) 54.0
14) 54.7
15) 55.0

Probability Density

1) 0.21
2) 0.39
3) 0.39
4) 0.61
5) 0.61
6) 0.61
7) 0.61
8) 0.61
9) 0.61
10) 0.64
11) 0.64
12) 0.53
13) 0.53
14) 0.14
15) 0.05

The probability density function I am using is:-
f(x) = 1 / Sqrt(2*PI*Sigma^2) * e^-[((x-mean)^2)/(2*Sigma^2)]

The first question is if the probability density at x = 52.7 is 0.21 (or in other words ( f(x)=0.21 ) what does the 0.21 mean in the context of my example of measuring the angles, what does the 0.21 mean? what does the density mean in the context of measuring an angle? 0.21 is a point that lies on my bell curve but what does that single point represent or mean?

The second question I wanted to ask is; I know that the area under the curve between two x values gives you a probability but my question is how can I interpret this probability in the context of measuring an angle? can I say that the probability of finding an angle between x = 53 and x = 54 is some value?

Any explanation in the context of my example will help me understand this.

I would really appreciate any help.

Thank you.

 

[scottdave]

Looking at the units (dimension) of the probability density function may give you some insight. What are the units of sigma?

 

[Stephen Tashi]

The most basic thing to understand is that "probability" is a mathematical concept rather than an empirical concept (such as specific measurements of something). When we apply probability theory to an empirical situation we often make assumptions about the relation between probability densities and actual measurements.

You are apparently assuming that your measurements are from a normal distribution. By using the mean and standard deviation calculated from a sample, you assume that the values for those statistics are the "true" values of the distribution. It should be obvious why this is an assumption. Probabilistic things don't always happen in a way that perfectly matches a theoretical distribution. For example, the average height of 100 randomly sampled people might not turn out to be the average height of the whole population. One branch of statistics deals with the problem of trying to estimate the parameters of the probability distribution for a population from calculations done only on a sample from that distribution.

That said, let's assume you have estimated the probability distribution correctly. Think of a probability density function in the way that you think of a mass density function. The mass density of an object can vary from point to point. The mass density of an object at one point might be 5 kilograms per cubic meter and yet the entire object might only weigh 1 kilogram. A point has no mass, but it can be assigned a mass density by imagining that we compute mass-per-unit-volume for a sequence of volumes that get smaller and smaller and are centered at the point.

If the probability density is $f(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2}}$ $e^{-\frac{(x - \mu)^2}{2\sigma^2}}$ then the value of $f(52.7)$ is a probability density, not a probability. (In your example, the density has units of probability-per-unit-radian or probability-per-unit-degree, depending on what unit you use to measure angles.)

To find the probability that $x$ is between 53 and 54, we compute $\int_{53}^{54} f(x) dx$ or use tables of the cumulative density to find that number. This is analogous to the way we find the mass of 1 cubic meter of an object when the mass density is varying over that cubic meter.

 

[tomtomtom1]

Looking at the units (dimension) of the probability density function may give you some insight. What are the units of sigma?

Hi ScottDave

The units are degrees.
But I still don't fully understand what this tells you in the context of measuring an angle.
All my units are in degrees and f(x)=0.21 when x = 52.7 degrees - what does the 0.21 tell you - all I know that the 0.21 tells you the probability density but what on Earth does that mean?

 

[tomtomtom1]

The most basic thing to understand is that "probability" is a mathematical concept rather than an empirical concept (such as specific measurements of something). When we apply probability theory to an empirical situation we often make assumptions about the relation between probability densities and actual measurements.

You are apparently assuming that your measurements are from a normal distribution. By using the mean and standard deviation calculated from a sample, you assume that the values for those statistics are the "true" values of the distribution. It should be obvious why this is an assumption. Probabilistic things don't always happen in a way that perfectly matches a theoretical distribution. For example, the average height of 100 randomly sampled people might not turn out to be the average height of the whole population. One branch of statistics deals with the problem of trying to estimate the parameters of the probability distribution for a population from calculations done only on a sample from that distribution.

That said, let's assume you have estimated the probability distribution correctly. Think of a probability density function in the way that you think of a mass density function. The mass density of an object can vary from point to point. The mass density of an object at one point might be 5 kilograms per cubic meter and yet the entire object might only weigh 1 kilogram. A point has no mass, but it can be assigned a mass density by imagining that we compute mass-per-unit-volume for a sequence of volumes that get smaller and smaller and are centered at the point.

If the probability density is $f(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2}}$ $e^{-\frac{(x - \mu)^2}{2\sigma^2}}$ then the value of $f(52.7)$ is a probability density, not a probability. (In your example, the density has units of probability-per-unit-radian or probability-per-unit-degree, depending on what unit you use to measure angles.)

To find the probability that $x$ is between 53 and 54, we compute $\int_{53}^{54} f(x) dx$ or use tables of the cumulative density to find that number. This is analogous to the way we find the mass of 1 cubic meter of an object when the mass density is varying over that cubic meter.

Hi Stephen Tashi

Thank you for your response.

Your analogy of a mass density function is something that I read elsewhere but I didn't fully understand it then.

All I have done is measured an angle 15 times in degrees.

You are correct I have assumed a normal distribution.

Using the probability density function i have calculated the probability density of each of my measured angles.

When f(52.7) is 0.21, I know that this is a probability density (thank you for explaining this), but can you explain what a probability density of 0.21 is in the context of my example which is measuring an angle.

I understand that to find a true probability I need an interval and then integrate the probability density function between that interval.
The bit that I don't fully understand is what does the probability density value tell you if the sample was based on measuring an angle?

Can you elaborate any further?

 

[scottdave]

Actually it is degrees^-1. The vertical unit is degrees^-1, when multiplied by the horizontal unit of degrees, you get a dimensionless "area".
The area under the entire curve should equal 1 (100% probability that some measurement will occur).

The value (at a particular point) does not have much meaning, but the "area" under the curve between two points does.
You could, for example, find the probability that the actual angle is off by 0.5° from the mean (53.11° to 54.11°), by integrating the function between those two values.
This could tell you something about the potential error which could be induced by this measurement.
If you go to this website http://davidmlane.com/hyperstat/z_table.html
for normal distributions, you can type in the mean and SD, then see the probability that between certain values.

 

[scottdave]

I also suggest reading the PF insights article about the Spectral Paradox https://www.physicsforums.com/insights/exploring-spectral-paradox/

It explains how density distributions work. Some of it can apply to your situation.

 

[Stephen Tashi]

When f(52.7) is 0.21, I know that this is a probability density (thank you for explaining this), but can you explain what a probability density of 0.21 is in the context of my example which is measuring an angle.

In the context of your experiment, the probability that your next measurement will be in the interval $[52.7 - a/2, 52.7 + a/2]$ is approximately $(0.21)(a)$, where $a$ is some small number. (The smaller $a$, the better the approximation.)

$(0.21)a$ is an approximation for $\int_{52.7 - a/2}^ {52.7 + a/2} f(x) \ dx$.

 

[tomtomtom1]

Actually it is degrees^-1. The vertical unit is degrees^-1, when multiplied by the horizontal unit of degrees, you get a dimensionless "area".
The area under the entire curve should equal 1 (100% probability that some measurement will occur).

The value (at a particular point) does not have much meaning, but the "area" under the curve between two points does.
You could, for example, find the probability that the actual angle is off by 0.5° from the mean (53.11° to 54.11°), by integrating the function between those two values.
This could tell you something about the potential error which could be induced by this measurement.
If you go to this website http://davidmlane.com/hyperstat/z_table.html
for normal distributions, you can type in the mean and SD, then see the probability that between certain values.

Scottdave

Thank you for elaborating, I really did get hung up on the value of f(x) of the probability density function in terms of what it meant. Knowing that the point on the curve is just the probability density and has little to no value in my example is something that I have now accepted and is beginning to sink in, I really appreciate your insight.

Your point about the area under the curve does have value is something that I already understood. However you stated:-

"You could, for example, find the probability that the actual angle is off by 0.5° from the mean (53.11° to 54.11°), by integrating the function between those two values."

I wanted to clarify this comment a little.

Using the site http://davidmlane.com/hyperstat/z_table.html (which is very useful so thank you) if I were to enter:-

Mean = 53.61
SD = 0.61
and integrate between 53.11 - 54.11 I get a probability of 0.5876 (58.76%).

To correctly interpret this probability value is it correct to say that there is a 58.76% probability that the next measurement of my angle could fall between 53.11 - 54.11 with a 1 degree of potential error?

Is it also correct to say that there is a 41.24% that the measurement will fall outside of 53.11 - 54.11?

Would you agree with this?

Finally when you integrate between 53.11 - 54.11 does the probability value include 53.11 and 54.11 or does the probability relate to greater than 53.11 and < 54.11?

Thanks Scott