# Mathematical statistics: two-dimensional random variables

• Jan 7th 2009, 08:30 AM
LaraSoft
Mathematical statistics: two-dimensional random variables
Boys, help a "blonde" to decide a task, please. I was tangled and does not know as correct to decide this task (Itwasntme).

The two-dimensional random variable $(X,Y)$ has a function of probability density $f\left( {x,y} \right) = \frac{a}
{{{\pi ^3}\left( {{x^2} + 3{y^2} + {x^2}{y^2} + 3} \right)}}$
.
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: $x=0$, $y=0$, $x=1$, $y=1$.
• Jan 7th 2009, 08:40 AM
HallsofIvy
Quote:

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 (Itwasntme).

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

Quote:

b) the function of a distribution $F\left( {x,y} \right)$;
Okay, that is equal to the indefinite integral
$\int\int f(x,y)dydx$

Quote:

c) the probability of hit of a casual point $\left( {X,Y} \right)$ in a square which is limited straight: $x=0$, $y=0$, $x=1$, $y=1$.
$\int_0^1\int_0^1 f(x,y) dy dx$