# Thread: Mathematical statistics: two-dimensional random variables

1. ## 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 .

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

2. 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 $\displaystyle (X,Y)$ has a function of probability density $\displaystyle f\left( {x,y} \right) = \frac{a} {{{\pi ^3}\left( {{x^2} + 3{y^2} + {x^2}{y^2} + 3} \right)}}$.
Find
a) the parametr $\displaystyle a$;
The requirement is that the integral over the entire plane,
$\displaystyle \int_{x=-\infty}^\infty\int_{y=-\infty}^\infty f(x,y)dydx= 1$
choose a so that integral is equal to one.

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

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