Density Function problem

In summary, the problem asks to find the mean number of screams on a roller coaster ride based on a given density function. The answer is 4.71238898038, which is equivalent to 3π/2. The units and length of the track are not given, so it is recommended to confirm with the teacher for further clarification.
  • #1
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Homework Statement



The density function for the number of times the riders scream on a roller coaster is given by ...

f(x) = [tex]\frac{1}{3\pi}[/tex](1 - cos(2x)) if 0 [tex]\leq[/tex] x [tex]\leq[/tex] 3[tex]\pi[/tex]
and 0 otherwise.

Find the mean number of screams over the course of the ride.

The answer: 4.71238898038

Homework Equations



None, unless integration is done by hand, in this case it doesn't have to be.

The Attempt at a Solution



I tried taking a several different integrals and never got close to that answer. I tried taking the integral of that function from 0 to 3[tex]\pi[/tex], I tried putting an x in front of the function, I had it read in degrees and radians, i took out the 1/3pi at the start...just anything I could think of and nothing was close to the answer. Help?
 
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  • #2
What are the units of the density function? [tex]\frac{\mbox{screams}}{\mbox{unit length}}?[/tex]

And over what length of track are you supposed to take it over?
 
Last edited:
  • #3
It's not given.
 
  • #4
The answer, 4.71238898038, is [tex]\frac{3\pi}{2}[/tex].

Does that help?
 
  • #5
the7joker7 said:
It's not given.

which?

edit:

Is it possible you were supposed to find the mean over the length of track from [tex]x\in \left[0,\frac{4}{3\pi}\right][/tex]?
 
Last edited:
  • #6
The units, and length.
 
  • #7
the7joker7 said:
The units, and length.

In that case, I think you should see if your teacher forgot something.
 

1. What is a density function problem?

A density function problem is a mathematical problem that involves finding the probability distribution of a continuous random variable. It is used to determine the likelihood of a certain value occurring within a given range.

2. How is a density function problem different from a discrete probability problem?

A density function problem deals with continuous random variables, while a discrete probability problem deals with discrete random variables. In a density function problem, the probability is described by a smooth curve, while in a discrete probability problem, the probability is described by a series of distinct points.

3. What is the formula for a density function?

The formula for a density function is f(x) = (1/σ√(2π)) * e^(-(x-μ)^2/2σ^2), where μ is the mean and σ is the standard deviation of the random variable.

4. How is a density function problem used in real life?

Density function problems are used in various fields such as physics, economics, and engineering. They can be used to model and predict the behavior of complex systems, such as the stock market or weather patterns. They are also used in risk analysis and decision-making processes.

5. What are some common applications of density function problems?

Some common applications of density function problems include calculating the probability of a certain stock price, predicting the probability of a natural disaster, and determining the optimal production level for a manufacturing process. They are also used in data analysis to identify patterns and make predictions.

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