Distribution of a normalized Gaussian vector If $X \in \mathbb{R}^n$ is a vector of independent standard Gaussian variables, what is the distribution of $\frac{X}{\left\lVert{X}\right\lVert}$?
How would this change if each $x_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$?
 A: If you calculate $\frac{X}{\left\lVert{X}\right\lVert}$, the only values that you will get is 1 and -1
This will be same for whatever mean and standardDeviation you set


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*Distribution X



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*Distribution $\frac{X}{\left\lVert{X}\right\lVert}$

If we use a vector of Normal Variables, we will get an array like this
    # Relationship of variables
    # 10000 Rows of Vector Size 1000. Each dimension being a normally distributed variable
    import scipy

    means=[x-50 for x in np.random.randint(100,size=(1000))]
    stdDevs=[x for x in np.random.randint(50,size=(1000))]
    dataVector=[]
    dataVectorMod=[]
    for curX in range(1000):
        x=scipy.stats.norm.rvs(loc=means[curX], scale=stdDevs[curX], size=10000, random_state=None)
        xMod=[y/abs(y) for y in x]
        dataVectorMod.append(xMod)
    dataVector=np.array(dataVector)
    dataVectorMod=np.array(dataVectorMod)
    dataVector=dataVector.T
    dataVectorMod=dataVectorMod.T
    print(dataVectorMod)



array([[ 1.,  1., -1., ..., -1., -1.,  1.],
       [ 1.,  1., -1., ..., -1., -1.,  1.],
       [ 1.,  1., -1., ...,  1., -1.,  1.],
       ...,
       [ 1.,  1., -1., ...,  1.,  1.,  1.],
       [ 1.,  1., -1., ...,  1., -1.,  1.],
       [ 1.,  1., -1., ..., -1.,  1., -1.]])

