Math Related Things

Relation between Density and Histrogram funcion

Given $X \sim f(x)$ , where $f(x)$ is the density function of $X$ , and the width of each bin $W$ , we can produce histrogram function $H(x)$ , by sampling variable $X$ $N$ times, resulting in $x1,x2...xN$ .
Then, there’s a conclusion: $H(x) \approx f(x) \times W \times N$ .

How to generate a random direction?

When doing academic research, I need to generate a random direction, i.e. it points to any direciton in an equalled probability. At first, I thought, assign each dimension a random value(Uniform Distribution) is sufficient. After second thought, I find out it’s not correct.
Let’s say, a N-dimension direction $D$ is defined by $D = (X_1, X_2...X_N)$ .

• If we just let $X_i$ be a random variable, i.e. $\forall i=1,2..N, X_i \sim U(-1,-1)$ , $D$ is NOT a random direction.

• To generate a truly random direction, we can let $\forall i=1,2..N, X_i \sim N(0,1)$ , where N(0,1) is a Normal Distribution with $u=0$ , $\theta=1$ , at this time, $D = (X_1, X_2...X_N)$ points to every direction in the same probability. And length of $D$ , $R = \sqrt{(X_1^2 + X_2^2 + ... + X_N^2)} \sim N(0,1)$