Table 6 Summary statistics of independent variables of the random-effect Tobit regression model Variables Obs Mean Std.
Since ([ {{a_{i}},{a_{i+1}}} ] subseteq I), it is a truncated random variable of the random variable ({X _{I}}), where (i=0,1,ldots,m-1).
If ξ J ⊆ ξ I and J ⊂ I, then we call the random variable ξ J a proper truncated random variable of the random variable ξ I, written as ξ J ⊂ ξ I.
If ({X _{J}}subseteq{X _{I}}) and (Jsubset I), then we say that the random variable ({X _{J}}) is a proper truncated random variable of the random variable ({X _{I}}), written as ({X_{J}}subset{X _{I}}).
Since [ a i, a j ] ⊆ I, we know that ξ [ a i, a j ] is a truncated random variable of the random variable ξ I, where 0 ⩽ i < j ⩽ m.
Since ([ {{a_{i}},{a_{j}}} ] subseteq I), we know that ({X _{ [ {{a_{i}},{a_{j}}} ] }}) is also a truncated random variable of the random variable ({X _{I}}), where (0leqslant i< jleqslant m).
If ξ J ∈ J ⊆ I is also a continuous random variable and its probability density function is p J : J → ( 0, ∞ ), p J ( t ) ≜ p I ( t ) ∫ J p I, then we call the random variable ξ J a truncated random variable of the random variable ξ I, written as ξ J ⊆ ξ I.
If ({X _{J}}in Jsubseteq I) is also a continuous random variable and its probability density function is {p_{J}}:Jrightarrow ( {0,infty} ),qquad {p_{J}} ( t ) triangleqfrac{{{p_{I}} ( t ) }}{{int_{J}{{p_{I}}}}}, (2) then we say that the random variable ({X _{J}}) is a truncated random variable of the random variable ({X _{I}}), written as ({X _{J}}subseteq{X _{I}}).
The Bienaymé formula states that the variance of the sum of uncorrelated random variables equals the sum of variances of the random variables [ 30].
The results include error bars of these variables and hierarchy impact of the random variables on the solution.
