We consider $Y=X{lambda}Z,;{lambda}>0$, where X and Z are independent random variables, and Y is the length biased distribution or the equilibrium distribution of X. The purpose of this paper is to consider the distribution of X or Y when the distribution of Z is given and the distribution of Z when the distribution of X or Y is given, In particular, we obtain that the necessary and sufficient conditions for X to be $X^{2}({upsilon});is;Z{sim}X^{2}(2);and;for;Z;to;be;X^{2}(1);is;X{sim}IG({mu},;{mu}^{2}/{lambda})$, where $IG({mu},;{mu}^{2}/{lambda})$ is two-parameter inverse Gaussian distribution. Also we show that X is smaller than Y in the reverse Laplace transform ratio order if and only if $X_{e}$ is smaller than $Y_{e}$ in the Laplace transform ratio order. Finally, we can get the results that if X is smaller than Y in the Laplace transform ratio order, then $Y_{L}$ is smaller than $X_{L}$ in the Laplace transform order, and that if X is smaller than Y in the reverse Laplace transform ratio order, then $_{mu}X_{L}$ is smaller than $_{ u}Y_{L}$ in the Laplace transform order.