What statistical processes and methods are used by geneticists/molecular biologists to know where one gene starts and one ends?
I know of only one naive approach to determining the boundaries of a gene : RACE-PCR. There are two kinds, 3' and 5' RACE, which allow to find the respective extremities.
The rationale is the following :
You perform a reverse transcription of the transcript of interest using a specific primer. At this step you have a specific single stranded cDNA.
Then you add a stretch of identical nucleotides called the homopolymeric tail in 5' of the cDNA.
Finally you perform a PCR using one specific primer and one universal primer that recognizes the homopolymeric tail. You can sequence your amplified cDNA and find where it is located in the genome with a 1 bp resolution.
For the 3'RACE, the concept is the same but the poly-A tail is used instead of generating it yourself with the terminal transferase.
See this paper for a detailed protocol :
Also, the corresponding wikipedia article gives you more details about what is happening at each step, but beware, there is an error : it is said that for the 5'RACE, the terminal transferase appends the homopolymeric tail in 3' while it appends it in 5'
There are various software in which you can enter your sequence (let's say the whole genome sequence) and it can identify for you the putative open reading frames (ORFs), i.e. the start codons and the stop codons. Then, by using these putative genes, you can do a sequence alignment by using BLAST and then, based on the scores you can confirm that those are really ORFs. As this being the statistical approach, you can then verify your results in the wet lab, like agrimaldi suggested.
If your goal is to define the boundaries of the transcription unit (the part of the DNA that is transcribed) the above answer is accurate, although many people merely use homology to cloned cDNAs rather than RACE reactions. This approach has the benefit of defining alternative splice forms at the same time.
If your goal is to define the "ends" of the gene, it can only be done empirically and functionally because control elements (boundaries, enhancers, etc) are impossible to recognize using informatics, and even if one finds enhancers, it's not certain that those enhancers are used with specific genes. Some genes can be millions of base pairs long, so have hundreds of others genes interspersed. The "gold standard" for defining the boundaries of genes is to rescue the loss-of-function phenotype of a mutation with a transgene that contains the gene of interest. If the DNA that is transformed back into an organism can recover the wild-type condition of a mutation of a gene, it is assumed that all of the important parts of that gene are within the transgene.
Generally speaking you sequence the genome and then search for clues. There are usually specific sequences preceding a gene that help the translational equipment know "hello this is where we begin" as well as regions where proteins can bind that are used to enhance or inhibit translation of the gene.
Computers can be programmed to search through the sequence and bring up possible candidates for people to look at more closely.