Relationship between these two probability mass functions If I have two different discrete distributions of random variables X and Y, such that their probability mass functions are related as follows: 
$P(X=x_i) = \lambda\frac{P (Y=x_i)}{x_i} $   
what can I infer from this equation? Any observations or interesting properties that you see based on this relation? 
What if,
  P($X=x_i$) = $\lambda\sqrt{\frac{P (Y=x_i)}{x_i} }$  
In both cases, $\lambda$ is a constant.
 A: if you rewrite the equation as $P(Y = x) = xP(X = x)/\lambda$, then the distribution of $Y$ is called the length-biased distribution for $X$. it arises, for example, if one has a bunch of sticks in a bag and reaches in and selects one at random - where the probability a particular stick is selected is proportional to its length. if the lengths of the sticks are realizations of the random variable $X$, the distribution of the length of the selected stick is that of $Y$.
A: I don't have enough reputation yet, or this would be a comment.
A random variable can have only one probability mass function, so it is not clear what you are asking.  Where do these equations come from?
A: First, you don't have the freedom to choose $\lambda$.  (Maybe you already realize that?)  In order for $P(X = x_i)$ to be a true probability distribution, you must have 
$$\lambda = \left(\sum_{x_i} \frac{P(Y = x_i)}{x_i}\right)^{-1}.$$
Given that, if $Y$ is geometric with success probability $p$, and $P(X = x_i) = \lambda P(Y = x_i)/x_i$ then $X$ does have the logarithmic distribution with parameter $q = 1-p$.
Here's why:  $P(Y = k) = (1-p)^{k-1}p$, for $k = 1, 2, \ldots$.  Thus 
$$P(X = k) = \lambda \frac{P(Y = k)}{k} = \lambda \frac{q^{k-1} p}{k} = \frac{\lambda p}{q} \frac{q^k}{k}.$$
The expression on the right is the logarithmic probability mass function with $\lambda = \frac{-q}{p \ln p}$.
