Htykuut

If $Ysimoperatorname{Gamma}(gamma,delta)$ and $Zsimoperatorname{Beta}(alpha,beta)$ then their density functions are, respectively,
$$
f_Y(y)=frac{delta^gamma}{Gamma(gamma)}y^{gamma-1}e^{-delta y},quad y>0,quadgamma>0,quaddelta>0
$$
and
$$
f_Z(z)=frac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)}z^{alpha-1}(1-z)^{beta-1},quad 0leq zleq 1,quadalpha>0,quadbeta>0.
$$
Consider $X_1$ and $X_2$ having $operatorname{Gamma}(a+b,1)$ and $operatorname{Beta}(a,b)$ distributions, respectively, where $a,b>0$. Assume that $X_1$ and $X_2$ are independent.

How do i find the marginal density functions of $Y_1 = X_1X_2$ and $Y_2 = X_1(1-X_2)$?

I know that the marginal density function can be derived from the joint density, but since the joint is not given, how do I create it?

Also how do I manipulate the gamma function? first time I have come across it.

Plugging in the definition, $X_1$ following $operatorname{Gamma}(a+b,1)$ means its density is

$$f_{X_1}(x_1) = frac1{ Gamma(a+b)}, x_1^{a+b-1} e^{-x_1} qquad text{for}~~ 0 < x_1 < infty $$

The density of $X_2$ is

$$f_{X_2}(x_2) = frac{ Gamma(a+b) }{ Gamma(a) Gamma(b) }, x_2^{a-1} (1 - x_2)^{b-1} qquad text{for}~~ 0<x_2<1$$

The fact that $X_1 perp X_2$ means their joint density is just the direct product

$$f_{X_1X_2}(x_1,, x_2) = frac1{ Gamma(a) Gamma(b) }, x_1^{a+b-1} e^{-x_1} , x_2^{a-1} (1 - x_2)^{b-1} qquad text{for}~~ begin{cases}
0<x_1<infty \
0<x_2<1 end{cases} tag*{Eq.(1)}$$

The 2-dim transformation is
$$begin{cases}
Y_1 = X_1 X_2 \
\
Y_2 = X_1 (1 - X_2)
end{cases} Longleftrightarrow begin{cases}
X_1 = Y_1 + Y_2 \
\
X_2 = dfrac{ Y_1 }{ Y_1 + Y_2}
end{cases} qquad text{where}~~ begin{cases}
0<y_1<infty \
0<y_2<infty end{cases}$$
with the Jacobian (of the inverse mapping) as
$$J = left| begin{matrix} dfrac{ partial x_1}{ partial y_1} & dfrac{ partial x_1}{ partial y_2} \
dfrac{ partial x_2}{ partial y_1} & dfrac{ partial x_2}{ partial y_2}end{matrix} right| = left| begin{matrix} 1 & 1 \
dfrac{ y_2 }{ (y_1 +y_2)^2 } & dfrac{ -y_1 }{ (y_1 +y_2)^2 } end{matrix} right| = frac{-1}{ y_1 + y_2 }$$
The transformed joint density for $Y_1$ and $Y_2$ is
begin{align}
f_{Y_1Y_2}( y_1 ,~y_2 ) &= |J| cdot f_{X_1X_2}( x_1,, x_2)Bigg|_{x_1 = y_1+y_2, ,,x_2 = frac{y_1}{y_1 + y_2}} qquad text{, plug in Eq.(1)}\
&= frac1{ y_1 + y_2} cdot frac1{ Gamma(a) Gamma(b) }, (y_1 + y_2)^{a+b-1} e^{-(y_1 + y_2)} , left(frac{y_1}{ y_1 + y_2}right)^{a-1} left(frac{y_2}{ y_1 + y_2} right)^{b-1} \
&= frac1{ Gamma(a) Gamma(b) }, y_1^{a-1} y_2^{b-1} e^{-(y_1 + y_2)} qquad text{for}~~0<y_1<infty ,~0<y_2<infty
end{align}

The marginal density of $Y_1$ can be obtained from the joint as
begin{align}
f_{Y_1}(y_1) &= int_{y_2 = 0}^{infty} f_{Y_1Y_2}( y_1 ,~y_2 ) ,mathrm{d}y_2 \
&= frac1{Gamma(a)} y_1^{a-1} e^{-y_1} int_{y_2 = 0}^{infty} frac1{Gamma(b)} y_2^{b-1} e^{-y_2} ,mathrm{d}y_2 qquad scriptsizetext{integral is just the kernel of Gamma distribution} \
&= frac1{Gamma(a)} y_1^{a-1} e^{-y_1}
end{align}
Thus one identifies the distribution of $Y_1$ as $operatorname{Gamma}(a,1)$.

Similarly, or noting the symmetry in the joint $f_{Y_1Y_2}( y_1 ,~y_2 )$, we have $Y_2$ follows $operatorname{Gamma}(b,1)$.