Mathematical statistics: two-dimensional random variables

**LaraSoft** · January 7th 2009, 07:30 AM

Boys, help a "blonde" to decide a task, please. I was tangled and does not know as correct to decide this task

.

The two-dimensional random variable $(X,Y)$ has a function of probability density $f\left( {x,y} \right) = \frac{a}<br /> {{{\pi ^3}\left( {{x^2} + 3{y^2} + {x^2}{y^2} + 3} \right)}}$ .
Find
a) the parametr $a$ ;
b) the function of a distribution $F\left( {x,y} \right)$ ;
c) the probability of hit of a casual point $\left( {X,Y} \right)$ in a square which is limited straight: $x=0$ , $y=0$ , $x=1$ , $y=1$ .

**HallsofIvy** · January 7th 2009, 07:40 AM

Originally Posted by LaraSoft

Boys, help a "blonde" to decide a task, please. I was tangled and does not know as correct to decide this task

.

The two-dimensional random variable $(X,Y)$ has a function of probability density $f\left( {x,y} \right) = \frac{a}<br /> {{{\pi ^3}\left( {{x^2} + 3{y^2} + {x^2}{y^2} + 3} \right)}}$ .
Find
a) the parametr $a$ ;

The requirement is that the integral over the entire plane,
$\int_{x=-\infty}^\infty\int_{y=-\infty}^\infty f(x,y)dydx= 1$
choose a so that integral is equal to one.

b) the function of a distribution $F\left( {x,y} \right)$ ;

Okay, that is equal to the indefinite integral
$\int\int f(x,y)dydx$

c) the probability of hit of a casual point $\left( {X,Y} \right)$ in a square which is limited straight: $x=0$ , $y=0$ , $x=1$ , $y=1$ .

$\int_0^1\int_0^1 f(x,y) dy dx$

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