# Statistics Continuous Distributions

• whitehorsey
In summary, the conversation discusses the potential disorientation of homing pigeons when a pair of coils is placed around them and a magnetic field is applied that reverses the Earth's field. It is noted that under these circumstances, the bird's initial flight direction, denoted by θ, is uniformly distributed over the interval [0, 2π]. The conversation then discusses finding the density for θ, sketching the graph of the density, and shading the area corresponding to the probability that a bird will orient within π/4 radians of home. This probability is found to be 1/8 when integrating the density over the appropriate region. It is also mentioned that if 10 birds are released independently, the expected value for the number
whitehorsey
1. If a pair of coils were placed around a homing pigeon and a magnetic field was applied that reverses the earth’s field, it is thought that the bird would be disoriented. Under these circumstances it is just as likely to fly in one direction as in any other. Let θ denote the direction in radians of the bird’s initial flight. θ is uniformly distributed over the interval [0, 2π].
(a) Find the density for θ.
(b) Sketch the graph of the density. The uniform distribution is sometimes called the rectangular distribution. Do you see why?
(c) Shade the area corresponding to the probability that a bird will orient within π/4 radians of home, and find this area using plane geometry.
(d) Find the probability that a bird will orient within π/4 radians of home by integrating the density over the appropriate region(s), compare your answer to that obtained in part (c).
(e) If 10 birds are released independently and at least 7 orient within π/4 radians of home, would you suspect that perhaps the coils are not disorienting the birds to the extent expected? Explain based on the probability of this occurring.

3. I haven't done the other parts because I'm stuck on finding the density for θ:
Based on the pdf formula, I got this but I don't know what the f(x) dx part should be
a. f(θ) = ∫2∏0 "f(x) dx?"

You are told what that density is: "θ is uniformly distributed over the interval [0, 2π]". Do you not understand what "uniform density" means?

HallsofIvy said:
You are told what that density is: "θ is uniformly distributed over the interval [0, 2π]". Do you not understand what "uniform density" means?

Oh! Just to make sure the equation would be like this:
f(θ) = 1/(2∏ - 0) = 1/2∏ 0 ≤ θ ≤ 2∏
0 otherwise

whitehorsey said:
Oh! Just to make sure the equation would be like this:
f(θ) = 1/(2∏ - 0) = 1/2∏ 0 ≤ θ ≤ 2∏
0 otherwise

Yes, that is the density function.

LCKurtz said:
Yes, that is the density function.

Thank You!

For c) Would the red part be the area I shade and calculate?

whitehorsey said:
For c) Would the red part be the area I shade and calculate?
View attachment 55301
It's 'within' π/4. So it's up to that error in either direction.

haruspex said:
It's 'within' π/4. So it's up to that error in either direction.

So these two parts?

Have you worked part (b) yet? You need to understand that to work part (c). The area in question is not shown in the pictures you are showing.

LCKurtz said:
Have you worked part (b) yet? You need to understand that to work part (c). The area in question is not shown in the pictures you are showing.
Hmm.. yes. I hadn't read part (b) - just assumed the area whitehorsey shaded was on the correct graph. Wrong assumption.

LCKurtz said:
Have you worked part (b) yet? You need to understand that to work part (c). The area in question is not shown in the pictures you are showing.

haruspex said:
Hmm.. yes. I hadn't read part (b) - just assumed the area ... Wrong assumption.

For part b I got this:

and for why its a rectangle distribution because all values in the range between 0 and 2π are equally likely.

Oh so I should be shading the attachment above like this:

whitehorsey said:
For part b I got this:
View attachment 55313
and for why its a rectangle distribution because all values in the range between 0 and 2π are equally likely.

Oh so I should be shading the attachment above like this:
View attachment 55312

That's better. To be complete your picture should indicate the "home" direction so you can tell whether or not you are within ##\pi/4## of home.

whitehorsey said:
I should be shading the attachment above like this:
View attachment 55312
Yes, except that you have fallen back into the range error in your original shading.

LCKurtz said:
That's better. To be complete your picture should indicate the "home" direction so you can tell whether or not you are within ##\pi/4## of home.

haruspex said:
Yes, except that you have fallen back into the range error in your original shading.

Ah okay! So I put back in the range and as a total I got .25. Is that correct?

whitehorsey said:
Ah okay! So I put back in the range and as a total I got .25. Is that correct?

Yes.

haruspex said:
Yes.

Thank You!

As for d, I took the integral of 1/2π from 0 to π/4 and got 1/8 but aren't part c and d's answers suppose to match up?

whitehorsey said:
As for d, I took the integral of 1/2π from 0 to π/4
That's the same mistake three times now

haruspex said:
That's the same mistake three times now

I'm so sorry! Lol it's the range! Okay fixed it! THANK YOU!

Lastly e. E(X)= np = 2.5?

whitehorsey said:
e. E(X)= np = 2.5?
Yes, but I assume you realize there's more to do.

haruspex said:
Yes, but I assume you realize there's more to do.

Ah yes the explanation. I was wondering how would the 2.5 prove that the coils are not disorienting the birds? I know E(X) stands for the population mean but I can't seem to grasp the concept to how it relates.

## 1. What are continuous distributions in statistics?

Continuous distributions in statistics refer to probability distributions that have an infinite range of possible values, as opposed to discrete distributions which have a finite number of possible values. These distributions are used to model continuous data, such as height or weight, and are described using a probability density function.

## 2. What is the difference between continuous and discrete data?

Continuous data is data that can take on any value within a given range, while discrete data is data that can only take on specific, separate values. For example, height is a continuous variable because it can take on any value within a range (e.g. 5.6 feet), while the number of siblings a person has is a discrete variable because it can only take on whole number values (e.g. 2 siblings).

## 3. How are continuous distributions graphically represented?

Continuous distributions are typically graphically represented using a line or curve on a graph, known as a probability density function (PDF). The area under the curve represents the probability of a data point falling within a specific range of values. The shape of the curve can vary depending on the distribution being used.

## 4. What is the central limit theorem and how does it relate to continuous distributions?

The central limit theorem states that the distribution of sample means of a population will approximate a normal distribution, regardless of the shape of the population's distribution. This is important for continuous distributions because it allows us to make inferences about a population based on a sample, assuming the sample is large enough.

## 5. What are some common examples of continuous distributions?

Some common examples of continuous distributions include the normal distribution, the exponential distribution, and the beta distribution. The normal distribution is often used to model data that follows a bell-shaped curve, while the exponential distribution is commonly used to model the time between events. The beta distribution is often used to model data that is limited to a specific range of values.

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