Fitting probability distribution to data 
I am trying to fit a model for the values plotted above. The explanatory variable represents amounts of compensation claim in an earthquake, and the response variable represents amounts of compensation awarded. Can someone tell me what probability distributions would be my possible options for these data? Thanks in advance.
 A: The relationship looks linear so you might consider a linear regression of  received on claimed. 
But I think you are going to have a problem with the distribution of residuals. There is almost a ceiling effect (very few people get more than they claim) and thus, if the residuals have 0 mean they probably won't be normally distributed.
It also looks like there will be some influential points, even on the log scale (e.g the $10,000,000 claim/award).
I would consider quantile regression models, which make no assumptions about the distribution of the residuals; also, interest may focus on the upper quantiles.
A: You have two parts to your answer: data and noise.
Someone could fit a line, or a spline to the mean of the data and if they were honest they would say something like "given this model, the mean tendency is ...".  
There is variation from the mean.  It looks like a 3 million dollars claim got 0.  If you had a line fit to predict the amount you might get a number like 3 million.  When someone has a claim for anything above 7.5 million dollars the departure from the linear fit collapses.  
First things first - I don't like your coordinates.  They should be log-log.  This would more clearly show the regions with the majority of the data.  Your y-axis should be the claim de-rating.  $$ y = claim - actual $$ or the log base 10 thereof.
Second - you want to account for variation.  You should be able to put a +/- 95% confidence interval around whatever someone asserts as the mean de-rating given a particular value of claim.  
Third - there is a clear difference between the 7.5 million dollar claim and the 3.0 million dollar claim.  Access to a variable like "how much did you pay your lawyer" or "which firm did you use" might allow clustering of the claims into higher and lower yield buckets.  You would then say "if you use x as lawyer and you agree to pay them y, then there is a 95% chance that if you claim z then you will actually get z-thisvalue."
A plot in the form requested (log-log, and transformed y-coordinate) is more likely to result in a resonable distribution.  Could you make it one that looks like this? (link)
Best of luck.  
PS: I bet a power law governs here.  (link)
PPS: And Peter is a genius.  If he suggests something and you can figure out how to do it - he is very likely to have given an insightful and informative answer.
A: Looks like a fairly linear relation to me. I would just start with a linear regression and inspect the residuals to see if I need to do any more, e.g. do something with the outlier claimed $\approx$ 7.5 milion, awarded $\approx$ 1 milion. 
A: Looks and sounds to me like these variables follow a gamma-distribution, basically. Following this idea, I simulated some data in R to emulate yours:
set.seed(1);y=rgamma(1000,.5);x=5*y+rgamma(1000,.5)+.3*rnorm(1000);plot(x,y)

I used shape parameter $k=.5$ for the random gamma distributions; $x$ is $y\cdot5$ for the strong linear relationship, plus a little random noise that is gamma-distributed (mostly to represent the group of people who claim something but get nothing) and normally-distributed (to make awards just a little higher or lower than claims).
