A 1D Random Walk Is An Example Of A Binomial Distribution

A 1 Dimensional Random Walk Is An Example Of A Binomail Distribution

A 1-dimensional random walk is an example of a binomial distribution, with a step either to the right or to the left being equally probable. Suppose out of N = 5 steps, x=3 steps are to the right with p = 0.5 and q = 0.5. Answer the following questions: Use (5) to find the number of steps to the right (of the total of N) steps. What is the probability that 3 steps are to the right? Calculate the mean number of steps to the right. Find the variance. What is the standard deviation?

Paper For Above instruction

The one-dimensional random walk is a fundamental concept in probability theory and stochastic processes, exemplifying the properties of the binomial distribution. In a simple random walk, at each step, an individual moves either to the right or to the left, each with equal probability, assuming p = q = 0.5. Specifically, this model is often used to demonstrate fundamental statistical concepts such as binomial probabilities, expected values, variances, and standard deviations. This paper explores these concepts within the context of a binomial distribution applied to a one-dimensional random walk with specified parameters.

Understanding the Binomial Distribution in Random Walks

The binomial distribution describes the probability of achieving a specific number of successes (in this case, steps to the right) in a fixed number of independent and identically distributed Bernoulli trials. The probability mass function (pmf) of the binomial distribution is given by:

P(X = x) = C(n, x)  p^x  q^{n-x}

where:

  • C(n, x) = n! / [x! * (n - x)!] is the binomial coefficient,
  • n is the total number of trials (steps),
  • x is the number of successes (steps to the right),
  • p is the probability of success on each trial, and
  • q = 1 - p is the probability of failure (step to the left).

Calculating the Probability of Exactly 3 Steps to the Right

Given N = 5 steps, with p = q = 0.5, and x = 3, the probability that exactly 3 steps are to the right is calculated as:

P(X = 3) = C(5, 3)  (0.5)^3  (0.5)^{2} = 10  0.125  0.25 = 10 * 0.03125 = 0.3125

This indicates there is a 31.25% chance that exactly three of the five steps will be to the right, consistent with the symmetric nature of the process.

Expected Value (Mean) of the Number of Steps to the Right

The expected number of successes (steps to the right) in a binomial distribution is determined by:

μ = n * p

Substituting the known values:

μ = 5 * 0.5 = 2.5

Thus, on average, we expect 2.5 steps to be to the right out of five.

Variance and Standard Deviation

The variance of a binomial distribution quantifies the spread or variability of the number of successes:

σ^2 = n  p  q

Calculating with the given parameters:

σ^2 = 5  0.5  0.5 = 5 * 0.25 = 1.25

The standard deviation, which is the square root of the variance, is:

σ = √1.25 ≈ 1.118

This measure indicates the typical deviation from the mean, reflecting the variability inherent in the random walk process.

Conclusion

In conclusion, the binomial distribution provides a robust framework for understanding the probabilistic behavior of a one-dimensional random walk. The specific calculations demonstrate the likelihood of various outcomes, with the probability of exactly three steps to the right being approximately 31.25%. The mean of 2.5 steps aligns with the expectation based on the probability of moving right, while the standard deviation of about 1.118 reflects the variability around this mean. These statistical insights are fundamental in analyzing stochastic processes and have applications across fields such as physics, finance, and biology.

References

  • Grinstead, C. M., & Snell, J. L. (2012). Introduction to Probability. American Mathematical Society.
  • Ross, S. M. (2014). A First Course in Probability. Pearson.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley.
  • DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics. Pearson.
  • Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer.
  • Durrett, R. (2010). Probability: Theory and Examples. Cambridge University Press.
  • Kendall, M., & Stuart, A. (1973). The Advanced Theory of Statistics, Volume 2. Charles Griffin & Company Ltd.
  • Gelman, A., et al. (2013). Bayesian Data Analysis. Chapman and Hall/CRC.
  • Billingsley, P. (1995). Probability and Measure. Wiley.

Related Assignments