Non central F distributions Let m n 2 N, a1 am 2 R, D Pm iD1 a2 i , and X1 Xm, Y1 Yn be i i d standard normal random variables Show that the random variable has the distribution density for y 0 The corresponding probability measure is called the non central Fm n distribution with non centrality parameter Hint Problem 9 10c Fm n a am m m (X a ) y j 1

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Question: Non-central F-distributions. Let m; n 2 N, a1; : : : ; am 2 R, D Pm iD1 a2 i , and X1; : :

Non-central F-distributions. Let m; n 2 N, a1; : : : ; am 2 R, D Pm iD1 a2 i , and X1; : : : ; Xm, Y1; : : : ; Yn be i.i.d. standard normal random variables. Show that the random variable

Fm.n;a....am m m (X; +a;) / y j=1

has the distribution density

image text in transcribed

for y 0. The corresponding probability measure is called the non-central Fm;n-distribution with non-centrality parameter . Hint: Problem 9.10c.

Fm.n;a....am m m (X; +a;) / y j=1

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