Intereting Posts

Is there a bounded continuous function $f$ s.t. a certain integral does not converge uniformly to zero?
Why is real exponentiation continuous in the base?
Do non-square matrices have eigenvalues?
integral involving invertible functions
How prove this inequality $(a^3+b^3+c^3)(ab+bc+ac)\ge 6abc(a^2+b^2+c^2-ab-bc-ac)$
Where is the flaw in my Continuum Hypothesis Proof?
An analogue of Hensel's lifting for Fibonacci numbers
Help proving exercise on sequences in Bartle's Elements
Adding a different constant to numerator and denominator
Summation notation problem
Combinatorial Proof with Summation Identity
Find $x$: $5\cos(x) = 2\sec(x)-3$
Integer lattice points on a sphere
Why are groups more important than semigroups?
Is there finest topology which makes given vector space into a topological vector space?

A particles moves in $\mathbb{R^2}$ started at the origin. At each stage $i (i = 1, 2, …)$, the particles would move, independently of all the stages before, one of the four directions North, East, South and West 1 unit, with probability $1 \over 4 $ each.Let $T_n$ be the distance from the origin just after $n$ steps. What is $E(T_n)$. I tried define 4 $1\times 2$matrice with only $1,0,-1$ in all entries. The use technique similar to simple random walk in 1 dimension and we know that for a $1\times 2$ matrice $(x,y)$, the distance from origin =$\sqrt {x^2+y^2}$ but i am not sure how to like the 1 D to 2 D. As in 2 D there are 4 possible direction.

- On Martingale betting system
- What is the average rotation angle needed to change the color of a sphere?
- Free throw interview question
- Symmetric Stable Distribution
- Expectation value of a product of an Ito integral and a function of a Brownian motion
- measurability question with regard to a stochastic process
- Probability Density Function Validity
- Sample variance derivation
- Probability a product of $n$ randomly chosen numbers from 1-9 is divisible by 10.
- If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.

An explanation is given here.

Using that formula you get for the 2D case for the distance from the origin just after n steps:

$$\sqrt{2n} \dfrac{\Gamma(\frac{3}{2})}{\Gamma(\frac{1}{2})}=\sqrt{\frac{n}{2}}$$

- Name a ring of 2 by 2 matrices where $a^3 = a$ and a belonging to this ring?
- Fourier cosine transform
- How to evaluate the series $1+\frac34+\frac{3\cdot5}{4\cdot8}+\frac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots$
- Mnemonics for linear algebra
- Pointwise vs. Uniform Convergence
- Is there a series to show $22\pi^4>2143\,$?
- Proof $ \int_0^\infty \frac{\cos(2\pi x^2)}{\cosh^2(\pi x)}dx=\frac 14$?
- Exact value of $\sum_{n=1}^\infty \frac{1}{n(n+k)(n+l)}$ for $k \in \Bbb{N}-\{0\}$ and $l \in \Bbb{N}-\{0,k\}$
- Different versions of Riesz Theorems
- A $\sin^n x$ integral
- PDF of product of variables?
- Curve enclosing the maximum area
- Is this a Delta Function? (and Delta as limit of Gaussian?)
- Right identity and Right inverse implies a group
- Inverse function of a polynomial and its derivative