Calculation of confidence interval of a population parameter (the range) Consider that $P$ is the water pressure coming out from a valve A. Let $P_{dif}$ be defined as the difference between the maximum and the minimum pressure of valve A: 
$$P_{\text{dif}}:= P_{\text{max}} - P_{\text{min}}$$
Now, what I want to do is to estimate $P_{\text{dif}}$. In order to do that, I take a number of water pressure samples from valve A. Let $S$ be a set of 3 measured samples: 
$$S = \{X_1 = 5, X_2 = 7, X_3 = 1\}$$ That is, $S$ contains 3 random samples drawn from the population. Therefore, I then say that $\hat{P}_{\text{dif}} = 7 – 1 = 6$.
First question: Considering a Gaussian distribution of the population parameter, how can I find the pdf, cdf of the parameter? 
Second question: Would a $95\%$ confidence interval (CI) of my estimation be defined as following?:
$$(pdf, mean) - 2\cdot (pdf, std) \leq (pdf, mean) \leq (pdf, mean) + 2\cdot (pdf, std) $$
where (pdf, mean) is the mean of the pdf and (pdf, std) its standard deviation.
If yes, how can I derive the (pdf, mean) and the (pdf, std)?
Edit:
As it seems quite difficult for me to find the CI of the range estimation, I am wondering whether I could use an alternative approach. whuber suggested that successive estimate should not depend strongly on the preceding ones. Say that I split my data samples into equal parts (e.g. 3) and I find one estimate for each of these sub-sets. Next, I find the 95% CI of the $\hat{P}_{dif}$ by that:
$$
mean - 2\cdot std \leq mean \leq mean + 2\cdot std
$$
where mean and std are the mean and std of the 3 different estimates, respectively. Would that be correct?
 A: This is difficult to answer, since you are using a strange terminology. But I take it (from the discussion in comments) That you are interested in the population range.  But then you cannot use a normal distribution model, since the population range in that case will be $(-\infty, \infty)$, and there is nothing to estimate. You need to model with a distribution which is concentrated on the range. 
The simplest such distribution is a uniform distribution on the range $(\theta_1, \theta_2)$, but is easy to construct others, like beta distributions or a restricted normal distribution. You must choose a model, but for this answer I will assume a uniform distribution. Then with your notation, it is easy to see that with probability one we have
$$
   \theta_1 \le P_{\text{min}} \le P_{\text{max}} \le \theta_2
$$
so that $P_{\text{dif}}= P_{\text{max}} - P_{\text{min}}\le \theta_2-\theta_1$.  This makes clear that the population range cannot be smaller than the sample range, so a confidence interval you have given in the post subtraction and adding two standard deviations is not correct. To find a confidence interval you can adopt the methods from Confidence interval for Uniform($\theta$, $\theta + a$).
