Suppose the distribution of Y , conditional on X = x, is N (x, x^2) and that the marginal distribution of X is uniform (0, 1). How do I prove that Y/X (division) and X are independent?
the joint density of (x,y) is f_XY(x,y)=1/[sqrt(2pi)x] exp( -(y-x)^2/2x^2 )
Let u =y/x and v=x. The Jacobian:
D(u,v)/D(x,y) = 1/v. Equivalently, D(x,y)/D(u,v) = v
Then,
f_UV(u,v) = v. f_XY(x(u,v),y(u,v) = v. f_XY ( v, uv) = 1/sqrt(2pi) exp(-1/2(u-1)^2)
Therefore, U and V are independent, V=X follows Unif(0,1) and U=Y/X follows N(1,1)
Given X~ U(0, 1) and Y ~ N(x, x^2), the cumulative distribution function (c.d.f.) of Y, F(y) is given by
F(y) = PHI(y/x-1)/sqrt(2pi),
where PHI(.) is a c.d.f. of the standard normal distribution.
Then, for U=Y/X, its c.d.f., F(u), is given
F(u) = PHI(u-1)/sqrt(2pi),
Now, let us consider a joint c.d.f. of (U, X), F(u, x):
F(u, x) = P(U
Proposition
En supposant que X suit la loi uniforme sur l’intervalle 📷et que 📷 suit la loi normale de moyenne x et de variance x² n la densité conjointe est donnée par 📷
📷 et 📷 lorsque X est uniforme sur 📷
Dans ce cas cette densité conjointe s’écrit sous la forme 📷, le développement et simplification permettent d’avoir 📷.
On peut maintenant trouver la densité conjointe de 📷 en faisant un changement de variables et le jacobien.
En posant x=u et y=uv
On calcule le déterminant de la matrice jacobienne J de la transformation inverse. (x,y) vers (u,v).
On obtient 📷
Donc le changement de densité donne 📷.
On observe que la densité conjointe marginale de 📷 c’est-à-dire de V est 📷 donc elle suit la loi normale de moyenne 1 et de variance 1
To prove this, one can start by defining a transformation Z=Y/X, Keeping X the same, i.e., X=X. This allows you to use a different pair of variables (Z, X) instead of the original pair (Y, X).
Then, show that the joint probability density function (PDF) of Z and X is equal to the product of their individual (marginal) PDFs. This would imply Z=Y/X is independent of X.
A question to all of you : can you open the addended excel file?
and does it appear in english language?
If not hwow to change MS - Excel=
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