Applications of stochastic processes What are stochastic processes? What are they used for? How can they be applied to real concepts? What is an example of a "stochastic process" problem?
 A: Stochastic processes are used everywhere - queuing theory (applied to communication networks among other things), statistical signal processing (adaptive filtering, estimation problems, RADAR, etc.), operations research, finance (see Shreve's Mathematical Finance text), etc. 
A stochastic process is simply a collection of random variables. Usually these random variables obey some sort of structural relation - for example, they may be related in time (time series) or space (such as wireless signal strengths in space), or through something like a Markov random field (used in machine learning). 
A: A stochastic process in loose terms is dynamics resulting from probabilistic fluctuations (like the spread of a disease in a population). They are used throughout condensed matter physics to get accurate descriptions of phenomena; they are used in stock market predictions; they are used in traffic simulations; they are used in epidemic modeling. A very easy example to understand is the susceptible-infected-recovered (SIR) model of disease transfer. I did a project for a course last year on epidemic models and comparing the deterministic and stochastic projections. You can find the PDF here.
A: A stochastic process is an ensemble of deterministic waveforms, or realizations, where each waveform is a function of time. Just as the random variable $X$ maps each outcome $\omega$ in sample space $\mathcal{S}$ to $\mathbb{R}$, the random process $X(t)$ maps each outcome to a deterministic function of time. If time is fixed, say, at $t_1$, the random process $X(t_1)$ is nothing but a random variable. 
Stochastic processes find tremendous applications in Finance and Electrical Engineering, esp. in Communications and Signal Processing. Noise in communication receivers, which is modeled as additive white Gaussian noise process, is one example of such a stochastic process.
