Density of the first k coordinates of a uniform random variableSuppose that X is distributed uniformly in the n-sphere n S n − 1 ⊂ R n . Then apparently the distribution of ( X 1 , … , X k ), the first k &lt; n coordinates of X has density p ( x 1 , … , x k ) with respect to Lebesgue measure in R k , moreover if r 2 = x 1 2 + ⋯ + x k 2 , then it is proportional to ( 1 − r 2 n ) ( n − k ) / 2 − 1 , if 0 ≤ r 2 ≤ n ,and otherwise is 0. I tried to compute this using the fact that ( X 1 , … , X k ) = d n ( g 1 , … , g k ) / g 1 2 + ⋯ + g n 2 , when g i are iid standard normal variables, but was unable to simplify the integrals. Does anyone know/can point me to a place where this density is derived?

Question

Density of the first k coordinates of a uniform random variableSuppose that X is distributed uniformly in the n-sphere       n              S              n      −      1        ⊂            R        n  . Then apparently the distribution of   (      X    1    ,  …  ,      X    k    ), the first   k  &amp;lt;  n coordinates of X has density   p  (      x    1    ,  …  ,      x    k    ) with respect to Lebesgue measure in             R        k  , moreover if       r    2    =      x    1    2    +  ⋯  +      x    k    2  , then it is proportional to            (      1      −                        r          2                n            )              (      n      −      k      )              /            2      −      1        ,    if     0  ≤      r    2    ≤  n  ,and otherwise is 0. I tried to compute this using the fact that   (      X    1    ,  …  ,      X    k    )                    =                    d                  n    (      g    1    ,  …  ,      g    k    )      /              g      1      2        +    ⋯    +          g      n      2      , when       g    i   are iid standard normal variables, but was unable to simplify the integrals. Does anyone know/can point me to a place where this density is derived?

muilasqk · Accepted Answer

Step 1Let   n  ≥  3 be an integer. Let   (      X    1    ,      X    2    ,  ⋯  ,      X    n    ) be a random vector with a uniform distribution on the sphere       S    n    (      n    )  =  {  (      x    1    ,      x    2    ,  ⋯  ,      x    n    )  ∈            R        n    :         x    1    2    +      x    2    2    +  ⋯  +      x    n    2    =  n  }Step 2Let   1  ≤  k  ≤  n  −  2 be an integer.   (      X    1    ,      X    2    ,  ⋯  ,      X    k    ) has the joint probability density                              f                                    X              1                        ,                          X              2                        ,            ⋯            ,                          X              k                                      (                  x          1                ,                  x          2                ,        ⋯        ,                  x          k                )                            =                  n                      −            n                          /                        2            +            1                                                Γ            (                          n              2                        )                                              π                              k                                  /                                2                                      Γ            (                                          n                −                k                            2                        )                          (        n        −                  x          1          2                −                  x          2          2                −        ⋯        −                  x          k          2                          )                                                    n                −                k                −                2                            2                                      ,                                                  (                  x          1                ,                  x          2                ,        ⋯        ,                  x          k                )        ∈                              R                    k                ,                 0        &amp;lt;                  x          1          2                +                  x          2          2                +        ⋯        +                  x          k          2                &amp;lt;        n        .

Density of the first k coordinates of a uniform random variable. Suppose that X is distributed uniformly in the n-sphere sqrtn S^{n-1} sub R^n.

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