Problem with expected value (Random Walk)

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  • #1
{ imp }
2
0

Homework Statement


Hello, I was reading Feynman's lectures on physics, and I'm having trouble following some deductions in the part about Probability. The random walk is a problem in which someone starts at x = 0 ant then takes a step forward (x = 1) or backward (x = -1) and after N steps de distance traveled is DN. I'll atach the images regarding the problem because it's kind of large.

I don't really understand how does he get to the part

DN2 = N

How is that?

Homework Equations


They are all on the images.


The Attempt at a Solution



I'm thinking about <DN-12> = N - 1, but I don't know why would that be.

I'm sorry if this doesn't go here or if my post is a little hard to read. Thanks in advance.
 

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  • #2
{ imp } said:

Homework Statement


Hello, I was reading Feynman's lectures on physics, and I'm having trouble following some deductions in the part about Probability. The random walk is a problem in which someone starts at x = 0 ant then takes a step forward (x = 1) or backward (x = -1) and after N steps de distance traveled is DN. I'll atach the images regarding the problem because it's kind of large.

I don't really understand how does he get to the part

DN2 = N

How is that?

Homework Equations


They are all on the images.


The Attempt at a Solution



I'm thinking about <DN-12> = N - 1, but I don't know why would that be.

I'm sorry if this doesn't go here or if my post is a little hard to read. Thanks in advance.

If you know some elementary probability theory it is easy. We have D = d1 + d2 + ... + dN, where di = distance moved in step i (di = -1 or +1 with probability 1/2 each). The random variables d1, d2, ... are _independent_. There is a basic theorem in probability that says Variance(sum) = sum(Variance), provided that the terms are independent. Thus, Var(D) = N*Var(d1), and Var(d1) = E(d1 - m1)^2, where m1 = E(d1) = mean of d1. Since m1 = 0 we have Var(d1) = E(d1^2) = (1/2)*(-1)^2 + (1/2)*(1^2) = 1. (Here, E = standard probabilistic notation for "expectation" = "mean"; you might prefer to use < > instead; that is, EX = <X>.)

RGV
 

Related to Problem with expected value (Random Walk)

1. What is a "random walk"?

A random walk is a mathematical concept that describes a path or trajectory where each step or movement is determined by random chance. It can be thought of as a series of steps taken in a random direction, often represented by a graph or plot.

2. What does "expected value" mean in the context of a random walk?

In the context of a random walk, expected value refers to the average or expected outcome of a series of random movements. It is calculated by multiplying the probability of each step by the value of that step, and then summing all of these values together.

3. Why is there a problem with expected value in a random walk?

The problem with expected value in a random walk is that it may not accurately predict the actual outcome of the walk. This is because the randomness of each step can lead to unexpected patterns or deviations from the expected value, making it difficult to accurately predict the overall outcome.

4. How is the problem with expected value addressed in real-world applications?

In real-world applications, the problem with expected value in a random walk is addressed by using statistical techniques such as Monte Carlo simulations. These simulations run multiple iterations of the random walk and calculate an average outcome, providing a more accurate estimate of the expected value.

5. What are some practical uses of random walks and expected value?

Random walks and expected value have various practical applications, including financial modeling, predicting stock prices, analyzing trends in data, and developing computer algorithms. They can also be used to study and understand real-world phenomena, such as the movement of particles in a gas or the spread of a disease in a population.

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