- #1
anaisabel
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- Homework Statement
- Find integral of delta function in problem
- Relevant Equations
- equation 1 from solution
Delta function to calculate density of probablity
- Thread starter anaisabel
- Start date
In summary, the conversation discusses an integral problem involving a crocodile's length and its maximum range. The length of the crocodile is not relevant to the solution, but rather the range at a specific angle is important. The integral is complicated by the fact that the probability of hitting the crocodile would be 1 if it was infinitely long and lying on the +x axis. The speaker has attempted to solve the integral using a change of variables, but is struggling with evaluating it. Another person in the conversation provides an example of how to change variables in a similar integral problem.
- #2
vela
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In the future, please type your post rather than posting images of your work. Your writing (to me at least) isn't the easiest to read.
In any case, what do you mean by "my teacher breaks the integral"? Also, where does the length of the crocodile enter into the solution?
In any case, what do you mean by "my teacher breaks the integral"? Also, where does the length of the crocodile enter into the solution?
- #3
anaisabel
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Sorry , my handwriting is a bit messy, but basically he integrates fron 0 to pi/4 and then from pi/4 to pi/2.
The length of the crocodile isn't really relevant, i just wrote the problem for some context. What i really want to know is how to solve that integral. I know you have to change variables because inside the dirac there is a function depending on theta, i just don't know how to do it. It is the calculus part that i can't figure out.
The length of the crocodile isn't really relevant, i just wrote the problem for some context. What i really want to know is how to solve that integral. I know you have to change variables because inside the dirac there is a function depending on theta, i just don't know how to do it. It is the calculus part that i can't figure out.
- #4
vela
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I'd say it has to do with the fact that the range is maximum when ##\theta = \pi/4##.
- #5
anaisabel
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Can you elaborate a bit more, i am not understanding very well.vela said:
- #6
vela
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Try calculating the projectile's range for ##\theta=\pi/6## and ##\theta=\pi/3##. You'll get the same distance. Do you see how that would complicate the evaluation of the original integral?
- #7
anaisabel
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I have forgotten a bit of my calculus, so I know the distance is the same, but if i don't know why it would complicate?.
- #8
vela
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I don't see how it couldn't be. If the crocodile were infinitely long and lying on the +x axis, for instance, the probability of hitting it would be 1.anaisabel said:
What have you tried to do in evaluating the integral?anaisabel said:
- #9
anaisabel
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In when comes to length of the cocrodile, I haven't gotten that far because I can't solve the integral. I have tried to solve it, and I understand now why you have to separate in two intervals, from 0 to pi/4 and pi/4 to pi/2, because when you perform tha change of variable and change the limits it would go from 0 to 0 if you didnt break into intervals. I understood that, but what I don't understand after. After I separate into intervals and perform the change of variable y like it is in my attempt, I don't know how to evaluate the integral, using the equation 1. The density is 2/pi, so is a constant, so what is the meaning of what is inside of delta?vela said:
- #10
vela
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The problem says ##\theta## is uniformly distributed, which is a continuous distribution. If you take the crocodile to be a point particle, you're asking for the probability that ##\theta## take on one or two particular values. When you have a continuous distribution, that probability would be 0. I'm not sure why you're using a delta function here or what your thinking is behind your integral.
Nevertheless, here's an example of how to change variables. Consider
$$\int f(x) \delta(x^2-a^2)\,dx,$$ where ##a>0##. Let's call the argument of the delta function ##g(x) = x^2-a^2##. The formula for the change of variables is
$$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{\lvert g'(x_i) \rvert},$$ where the sum is over all the roots of ##g##. In this example, ##g(x)=0## at ##x=a## and ##x=-a##, so we get
$$\int f(x) \delta(x^2-a^2)\,dx = \int f(x)\left[\frac{\delta(x-a)}{2a} + \frac{\delta(x-(-a))}{2a}\right]\,dx = \frac{f(a)+f(-a)}{2a}.$$
Nevertheless, here's an example of how to change variables. Consider
$$\int f(x) \delta(x^2-a^2)\,dx,$$ where ##a>0##. Let's call the argument of the delta function ##g(x) = x^2-a^2##. The formula for the change of variables is
$$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{\lvert g'(x_i) \rvert},$$ where the sum is over all the roots of ##g##. In this example, ##g(x)=0## at ##x=a## and ##x=-a##, so we get
$$\int f(x) \delta(x^2-a^2)\,dx = \int f(x)\left[\frac{\delta(x-a)}{2a} + \frac{\delta(x-(-a))}{2a}\right]\,dx = \frac{f(a)+f(-a)}{2a}.$$
FAQ: Delta function to calculate density of probablity
1. What is a delta function and how is it used to calculate density of probability?
A delta function, denoted as δ(x), is a mathematical function that is commonly used in physics and engineering to represent an impulse or spike of a certain magnitude at a specific point. In terms of calculating density of probability, the delta function can be used to represent the probability of an event occurring at a specific value of a continuous random variable.
2. How is the delta function related to the concept of probability density?
The delta function is directly related to the concept of probability density, as it is used to represent the probability density function (PDF) of a continuous random variable. The area under the delta function curve represents the probability of an event occurring at a specific value, and the integral of the delta function over a range of values gives the probability of the event occurring within that range.
3. Can the delta function be used to calculate the probability of a continuous random variable taking on a specific value?
Yes, the delta function can be used to calculate the probability of a continuous random variable taking on a specific value. This is because the delta function has a value of zero everywhere except at the specific value, where it has a value of infinity. When integrated over a small range around the specific value, the delta function gives a finite probability for the event occurring at that value.
4. Are there any limitations to using the delta function to calculate density of probability?
One limitation of using the delta function to calculate density of probability is that it can only be used for continuous random variables. It cannot be used for discrete random variables, as they have a finite number of possible outcomes. Additionally, the delta function can only be used for events that have a probability of occurring at a single point, and cannot be used for events that have a probability of occurring over a range of values.
5. How is the delta function related to other probability distributions?
The delta function is a special case of the Dirac delta function, which is a more general mathematical concept used in various fields of science and engineering. The Dirac delta function can be used to represent other probability distributions, such as the Gaussian distribution, by convolving the delta function with the desired distribution. This allows for the use of the delta function in a wider range of applications beyond just calculating density of probability.