Introduction to Probability for Data Science
&quot;A random process is a function indexed by a random key.&quot;<p>Not just wrong, wildly bad nonsense.<p>Go get some data.  Now you have the value of a <i>random variable</i>.<p>We don&#x27;t get clear on just what <i>random</i> means, and in <i>random variable</i> we do not assume some element of not knowing.  In particular <i>truly random</i> is nonsense.<p>Suppose we have a non-empty set I and for each i in I we have a random variable X_i (using TeX notation for a subscript).  Then the I and the set of all X_i is a <i>random process</i> or a <i>stochastic process</i>.  We might write (X_i, I) or some such notation.<p>Commonly the set I is an interval subset of the real line and denotes time.  Set I might be half of the real line or all of it or just some interval, e.g., [0,1].<p>The set I might be just the numbers<p>[1, 2, 3, 4, 5, 6}<p>for, say, playing with dice with the usual six sides.<p>I might be the integers in [1, 52] for considering card games.<p>But the set I might be all the points on the surface of a sphere for considering, say, the weather, maybe the oceans, etc.<p>The set I might be all pairs (t, x, y, z) where t is a real number denoting time and the other three are coordinates in ordinary 3-space.<p>A random variable can also be considered a function with domain a <i>probability space</i> O.  So for random variable Y, for each w in O, Y(w) is the value of the random variable Y at <i>sample</i> w. Right, the usual notation has capital Greek omega for O and lower case Greek omega for w.<p>Then for a particular w and stochastic process X with index set I, all the X_t(w) as t varies is a <i>sample path</i> of the process X.  E.g., a plot of the stock marked DJI for yesterday is part of such a sample path.  So, with stochastic processes, what we observe are sample paths.<p>That&#x27;s a start on stochastic processes.  Going deep into the field gets to be difficult quickly.  Just quickly, look for names Kolmogorov, Dynkin, Doob, Ito, Shiryaev, Skorokhod, Rockafellar, Cinlar, Strook, Varadhan, Mckean, Blumenthal, Getoor, Fleming, Bertsekas, Karatzas, Shreve, Neveu, Tulcea(s).<p>For some of the <i>flavor</i> of probability theory and stochastic processes, see the article on <i>liftings</i> at<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Lifting_theory" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Lifting_theory</a><p>I had the main book on liftings, I&#x27;d gotten for $1 at a used book store (not a big seller) but lost it in a recent move.