Pair Of Random Variables Research Articles

A Diagonal Expansion for the 2 Dirichlet Probability Density Function

Let $X_1 $ and $X_2 $ be a pair of random variables with a joint density $p( {x_1 ,x_2 } )$, and marginal densities $p_1 ( {x_1 } )$ and $p_2 ( {x_2 } )$ respectively. If $\{ {\theta _n^{( 1 )} ( {x_1 } )} \}$ and $\{ {\theta _n^{( 2 )} ( {x_2 } )} \}$ are two sets of orthonormal polynomials, $n = 0,1,2, \cdots $ , associated with the weight functions $p_1 ( {x_1 } )$ and $p_2 ( {x_2 } )$ respectively, the diagonal expansion of Barrett and Lampard for $p( {x_1 ,x_2 } )$, if such a series exists, is given by \[ p( {x_1 ,x_2 } ) = p_1 ( {x_1 } )p_2 ( {x_2 } )\sum\limits_{n = 0}^\infty {a_n } \theta _n^{( 1 )} ( {x_1 } )\theta _n^{( 2 )} ( {x_2 } ) \] the coefficients $a_n $ being independent of both $x_1 $ and $x_2 $.Many examples of the Barrett–Lampard diagonal expansion are known. Another new example is given in this paper for the 2-variate Dirichlet probability density function defined as \[ p\left( {x_1 ,x_2 } \right) = \frac{{\Gamma \left( {\nu _1 + \nu _2 + \nu _3 } \right)}}{{\Gamma \left( {\nu _1 } ...

A Random Interval Filling Problem

Consider an initial real interval $\lbrack 0, x\rbrack, x > 0$, and place in it a random subinterval $I(x)$ defined by a pair of random variables $(U_x, V_x)$; the former being the length of $I(x)$ and the latter its lower boundary point. The set $\lbrack 0, x\rbrack - I(x)$ consists of two intervals of lengths $x_1$ and $x_2$, in which there are in turn placed random subintervals $I(x_1)$ and $I(x_2)$ defined by pairs of random variables $(U_{x_1}, V_{x_1})$ and $(U_{x_2}, V_{x_2})$. The process of placing random subintervals in $\lbrack 0, x\rbrack$ is thus continued. Under the assumptions that the subintervals cannot overlap, and that their lengths are uniformly bounded away from zero, the procedure must terminate after a finite number of steps. Let $N(b, x)$ denote the number of subintervals of $\lbrack 0, x\rbrack$ of length at least $b$, in the terminal state. The asymptotic behaviour of the moments of $N(b, x)$ is here studied as $x \rightarrow \infty$. It is shown that under fairly general conditions the mean approaches a linear function of $x$ at the rate $x^{-n}$, for any integer $n > 0$. Under the further condition that $V_x$ is a family of uniform distributions the exact form of the linear relation is determined. In the last section it is indicated how this result can be extended to some more general distributions. A similar but less precise result is proved for the higher moments, the convergence rate $x^{-n}$ not being established for this case.