Get distribution curve characteristics by selecting peak point? 
I have no math or stats background, so this may be a real dumbo question.
I'm developing a time estimation app which would let user enter her worst case and best case estimates, and then select a probability curve which she feels looks realistic.  
I needed to draw some kind of curve that sums up to 1, so I came up with a normal distribution. I'm not sure that it captures how we give time estimates but this is not what question is about.
I want user to be able to move finger on the chart, so the curve would bend accordingly. I assume it would feel nice if curve's peak followed the finger, but then I have the following question: is it possible to get normal distribution curve by arbitrary peak point? If not, which curve is good for this?
UPDATE:
I see that I can obtain μ from x location.
Can I get σ for an arbitrary y though (combined with known μ)?
 A: Because normal distributions are symmetric, and it's natural to think of "best" and "worst" cases symmetrically (e.g., the chance of the best case is no greater than 5% and the chance of the worst case is no less than the same percent), they are not likely to work well in this application.
In effect you have three degrees of freedom to specify a distribution.  Pick any family that requires three parameters and is suited to distributions of "time".  If you are estimating time to complete a task, people often have used scaled Beta distributions for this purpose.  Another good choice is the Weibull distribution, which often arises in such applications, and possibly the Gamma distribution.  The latter two require only two parameters, but many people have introduced a third location parameter that shifts the distribution (usually to the right), so-called "three-parameter Weibull" and "three-parameter Gamma".  You can also embed both these distributions into a larger three-parameter family that includes the normal, called a "generalized Gamma".
Typically some latitude is allowed for the time to exceed the "worst" case, so you would interpret the worst case as, say, somewhere between the 80th and 99th percentile of the distribution rather than it maximum (and similarly, mutatis mutandis, for the best case).  Thus, whatever family you pick, you need to be able to find its usual parameters (such as $\alpha$, $\beta$, and the scale factor for the scaled Beta distribution), given the values of a lower percentile (the best case), an upper percentile (the worst case), and the mode.  This can always be done (provided the percentiles and modes can be modeled by the chosen distributional family at all), but the amount of coding work--and the details of the mathematics--will vary with the distributional family.  Because this can challenge one's numerical programming skills, many applications use approximations, often gross ones (given the qualitative nature of the elicited information).  A common one occurs in task duration modeling, where scaled Beta distributions are estimated by means of formulas that are not exactly right but are extremely simple (they are linear combinations of the three values).
A: The normal curve is not going to work for you:


*

*it is not constrained by a minimum/maximum

*it is symmetric around the peak

*it has a very restrictive shape


There are loads of other statistical distributions that would probably fit your needs better. I would especially look at distributions confined to the 0-1 interval (and then rescaled to your min-max interval) such as the Beta distribution. The "peak" of the distribution is called "mode" in stat-speak. The formula for the mode is quite straightforward for the Beta. 
Note, however, that this distribution has two parameters, and you probably want to allow some way of changing both.
