Answer :
Expression would be,
[tex]E[r_n^2] = \sum \ x^2 * (n choose x) * (1/4)^x * (3/4)^(n-x)[/tex]
What is Unrestricted symmetric ?
The unrestricted, one-dimensional, symmetric random walk represented as Pascal's triangle.
In a two-dimensional unrestricted symmetric random walk, the particle moves in one of four directions (up, down, left, or right) with equal probability at each step. The distance of the particle from the origin after the nth step, [tex]r_n,[/tex] is given by the square root of the sum of the squared x and y coordinates of the particle at that time.
The expected value of[tex]r_n^2[/tex], denoted as [tex]E[r_n^2][/tex], is given by:
[tex]E[r_n^2] = \sum\ P(r_n^2 = x^2) * x^2[/tex]
where the sum is taken over all possible values of [tex]x^2[/tex].
Since the particle moves with equal probability in each of the four directions, we can calculate the probability that the particle is at a particular x and y coordinate using the binomial distribution:
[tex]P(x, y) = (n \ choose\ x) * (1/4)^x * (3/4)^(n-x)[/tex]
Substituting this expression into the formula for [tex]E[r_n^2][/tex] and summing over all possible values of x and y, we get:
[tex]E[r_n^2] = \sum \ x^2 * (n choose x) * (1/4)^x * (3/4)^(n-x)[/tex]
To know more about Unrestricted symmetric visit,
https://brainly.com/question/29409984
#SPJ4
