I´m trying to understand the notion of a stochastic process so I came with the following problem. Let $X$ be a standard normally distributed random variable and form the stochastic process $$Y_{t}=X+2t.$$

For this particular example, I have two questions:

1.- How can I describe the simple paths of the process?

2.- What is the probability that $Y_{t}=0$ for some $t\in \mathbb{N}$?

I´ve never came across with particular examples of stochastic processes, so I hope some illumination with this definitions.

Thank you.


1 Answer 1


When talking about stochastic processes you should always specify the time (or "index") set: $\mathbb R,$ $\mathbb R^+$, $\mathbb Z,$ $\mathbb N,$ or what?

Assuming you mean $\mathbb N$, your sample paths are arithmetic sequences increasing 2 at a time, starting at $Y_1 = 2+X$ where $X\sim N(0,1)$. For each $t\in\mathbb N$, $P(Y_t=0) = 0$, and by countable additivity, $P(\exists t\in\mathbb N : Y_t=0) = 0$. But if your time set is $\mathbb R$, of course for $t=-X/2$ you will have $Y_t=0$.

One way to think about stochastic processes is that they are collections of random variables, indexed by $t$; another is that are random functions of $t$. The latter is, I think, easier for beginners to think about, and you should develop some intuition and comfort level working with both points of view.

  • $\begingroup$ Is it correct to equate a deterministic variable to a random variable, $t=-X/2$? $\endgroup$
    – Deep
    Commented Sep 21, 2017 at 13:25
  • $\begingroup$ What does "correct" mean here. In the context of the particular example where the sample paths are functions $t\mapsto X+2t$, there is nothing wrong with saying that at time $t= -X/2$ the function takes the value 0. Even random lines with slope 2 cross the $x$-axis somethere. $\endgroup$ Commented Sep 21, 2017 at 13:48

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