How to find the value of $m$ given $f_{X,Y}(x,y)$

Question

given $(X,Y)$ random variable 2D continuously with the density function $$f_{X,Y}(x,y)=\begin{cases}\frac{1}{{30}}\cdot x^{m}\cdot ye^{-y} & x \in [0,y] \ , \ y>0 \\ 0 & \textrm{otherwise} \end{cases}$$

I need to find the value of $m$

I know I should use the properties of the density function $$1=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)dxdy$$ $$1=\int_{0}^{y}\int_{-\infty}^{\infty}\frac{1}{{30}}\cdot x^{m}\cdot ye^{-y}dxdy$$

I dont understand how to determine the limits of $x$ from the given about it.

Answer 1

In the computation of $\displaystyle\iint_{\Bbb R^2}f(x,y)\; dx\; dy$ you can use Fubini and

• either let $y$ run in the interval $[0,\infty)$, and then $x\in[0,y]$,
• or let $x$ run in the interval $[0,\infty)$, and then $y\in[x,\infty)$.

The two ways are: $$ \begin{aligned} 1 &= \int_{y\in[0,\infty)}dy\int_{x\in[0,y]} \frac 1{30}x^m\; y\; e^{-y}\; dx \\ &= \int_{y\in[0,\infty)}dy\cdot \frac 1{30}\cdot\frac {y^{m+1}}{m+1}y\; e^{-y}\\ &= \frac 1{30(m+1)}\cdot \Gamma(m+3) \qquad\text{ and if $m$ is integer} \\ &= \frac 1{30(m+1)}\cdot (m+2)! \\ &=\frac 1{30}(m+2)\cdot m! \ , \\[4mm] 1 &= \int_{x\in[0,\infty)}dx\int_{y\in[x,\infty)} \frac 1{30}x^m\; y\; e^{-y}\; dy \\ &= \int_{x\in[0,\infty)}dx\cdot \frac 1{30}x^m\; (x+1)\; e^{-x} \\ &=\frac 1{30}\cdot(\Gamma(m+2)+\Gamma(m+1)) \qquad\text{ and if $m$ is integer} \\ &= \frac 1{30}\cdot ((m+1)!+m!) \\ &=\frac 1{30}\cdot m!\cdot((m+1)+1) \\ &=\frac 1{30}(m+2)\cdot m! \ . \end{aligned} $$ For $m=3$ we have indeed equality.

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The equation to be solved $(m+2)\Gamma(m+1)=30$ has a unique solution $m>0$ since the L.H.S. is an increasing function of $m$.

But note that the $\Gamma$ function comes with "its own caprice", we find for instance...

? plot(m=-.99, -0.9, (m+2)*gamma(m+1) - 30) 

70.426911 "''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''|
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          -----------------"x_--------------------------------------------
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          |                                """"xxxx____                  |
-19.53514 |............................................""""""xxxxxxxx_____
          -0.99                                                       -0.9
? solve(m=-.99, -0.9, (m+2)*gamma(m+1) - 30) 
%21 = -0.96617414943112079757353172624510260510