Elicit a Proper informative priors I need help with eliciting,deriving and plotting two proper informative prior.
One with mean and standard deviation
Another with LQ, UQ and Median
Is the first one using a normal distribution while the second uses a scaled beta distribution? I'm not honestly very good at priors, so any help would be very appreciated.
 A: I'm answering on the premise that this is a homework question. Disclosure: I'm not a Bayesian, but I'm familiar with some concepts.
From comments, you appear to want to elicit a prior that describes a fraction. You seem to be asking, what sort of prior is conceptually sound for this? You seem like you may be debating between a normal prior and a beta prior.
The normal distribution has support of -$\infty$ to $\infty$. So, in principle, assuming a normal prior could mean that you have fractions less than zero or over 1. That might not be substantively sound.
The beta distribution has support (0, 1), i.e. 0 to 1 exclusive of 0 and 1. If you Google, you should find that it's often used as a prior for fractions. It has two key parameters, $\alpha$ and $\beta$. If you had $\alpha + \beta$ trials and you got $\alpha$ successes, you can derive the appropriate parameters for a beta distribution expressing that as a probability.
Imagine you asked 10 patients for their preference, and 3 said they'd prefer the treatment, i.e. 30% of the sample prefer the treatment, but it's a small sample, so we're not sure the 'true' proportion is 30% (clearly I'm a frequentist). That's a beta distribution with $\alpha = 3$ and $\beta = 7$, and its cumulative distribution function looks like this:

If you had instead asked 1000 patients for their preference, and 300 said they'd prefer the treatment, you have $\alpha = 300$ and $\beta = 700$, you still think that 30% of patients prefer the treatment, but you're a lot more certain about that value, as illustrated by this very different CDF:

If you want to make similar plots, you can just go to Wolfram Alpha and tell it, for example, "plot a beta distribution with a = 300 and b = 700".
I'm not sure how you are eliciting priors from experts on this. In principle, if you asked a statistician who had studied this area for a prior, she might respond, "I think the probability of choosing T follows a beta distribution with $\alpha = 3$ and $\beta = 7$, and unlike Mr. Ng, I'm a real Bayesian." It doesn't seem likely you'd get this sort of response. I rather suspect you would be asking a bunch of experts, maybe 10, if they think that the average patient would prefer C or T. Thus, you would be able to parameterize your prior according to how many respond C or T. If they give a more complex response, then I guess you will have to ask them to choose one or the other (or bribe them, or threaten them, or whatever).
I'm not familiar with "scaled" beta distributions. However, I hope this gets you on the right track to think about how and why you might use the beta distribution as a prior.
