How will studying "stochastic processes" help me as a statistician? I wish to decide if I should take a course called "INTRODUCTION TO STOCHASTIC PROCESSES" which will be held next semester in my University.
I asked the lecturer how studying such a course would help me as a statistician, he said that since he comes from probability, he knows very little of statistics and doesn't know how to answer my question.
I can make an un-educated guess that stochastic processes are important in statistics.  But I am also curious to know how.
That is, in what fields/methods, will basic understanding in "stochastic processes" will help me do better statistics?
 A: You need to be careful how you ask this question. Since you could substitute almost anything in place of stochastic processes and it would still be potentially useful. For example, a course in biology could help with biological statistical consultancy since you know more biology!
I presume that you have a choice of modules that you can take, and you need to pick $n$ of them. The real question is what modules should I pick (that question probably isn't appropriate for this site!)
To answer your question, you are still very early in your career and at this moment in time you should try to get a wide selection of courses under your belt. Furthermore, if you are planning a career in academia then some more mathematical courses, like stochastic processes would be useful.
A: A deep understanding of survival analysis requires knowledge of counting processes, martingales, Cox processes... See e.g. Odd O. Aalen, Ørnulf Borgan, Håkon K. Gjessing. Survival and event history analysis: a process point of view. Springer, 2008. ISBN 9780387202877 
Having said that, many applied statisticians (including me) use survival analysis without any understanding of stochastic processes. I'm not likely to make any advances to the theory though.
A: The short answer probably is that all observable processes, which we may want to analyze with statistical tools, are stochastic processes, that is, they contain some element of randomness. The course will probably teach you the mathematics behind these stochastic processes, e. g. distribution functions, which will allow you to grasp the function of your statistical tools. 
I think you can compare it with an automobile: As you can drive your car without understanding the engineering behind it, and without theoretical knowledge about the dynamics of your car on the road, you can apply statistical tools to your data without understanding how these tools work, as long as you understand the output. This will probably be good enough if you want to do basic statistics with well behaved data. But if you really want to get the most out of your car, to see where it's limits are, you need knowledge about the engineering, the dynamics of your car on roads and in curves and so on. And if you want to get the most out of your data with the help of your statistical tools, you need to understand how data generation can be modeled, how tests are devised and what the assumptions behind your tests are to be able to see where those assumptions are not valid.
A: Just for the sake of completeness, an IID sequence of random variables is also a stochastic process (a very simple one).
A: Stochastic processes underlie many ideas in statistics such as time series, markov chains, markov processes, bayesian estimation algorithms (e.g., Metropolis-Hastings) etc. Thus, a study of stochastic processes will be useful in two ways:

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*Enable you to develop models for situations of interest to you.
An exposure to such a course, may enable you to identify a standard stochastic process that works given your problem context. You can then modify the model as needed to accommodate the idiosyncrasies of your specific context.


*Enable you to better understand the nuances of the statistical methodology that uses stochastic processes.
There are several key ideas in stochastic processes such as convergence, stationarity that play an important role when we want to analyze a stochastic process. It is my belief that a course in stochastic process will let you appreciate better the need for caring about these issues and why they are important.
Can you be a statistician without taking a course in stochastic processes? Sure. You can always  use the software that is available to perform whatever statistical analysis you want. However, a basic understanding of stochastic processes is very helpful in order to make a correct choice of methodology, in order to understand what is really happening in the black box etc. Obviously, you will not be able to contribute to the theory of stochastic processes with a basic course but in my opinion it will make you a better statistician. My general rule of thumb for coursework: The more advanced course you take the better off you will be in the long-run.
By way of analogy: You can perform a t-test without knowing any probability theory or  statistics testing methodology. But, a knowledge of probability theory and statistical testing methodology is extremely useful in understanding the output correctly and in choosing the correct statistical test.
A: In medical statistics, you need stochastic processes to calculate how to adjust significance levels when stopping a clinical trial early. In fact, the whole area of monitoring clinical trials as emerging evidence points to one hypothesis or another, is based on the theory of stochastic processes. So yes, this course is a win.
A: Other areas of application for stochastic processes: (1) Asymptotic theory: This builds on PeterR's comment about an IID sequence.  Law of large numbers and central limit theorem results require an understanding of stochastic processes. This is so fundamental in so many areas of application that I am inclined to say that anyone with a graduate degree in stats or a field that uses sampling or frequentist inference ought to have key stochastic processes results under their belt.  (2) Structural equation modeling for causal inference a la Judea Pearl: Analyzing directed acyclic graphs (DAGs) of causal processes requires some handle of stochastic process theory. 
