Lect 5 - Random Variables (Contd.)
00:00 Start
01:00 Brief review of CDFs from previous lecture (Continuous and Discrete RVs)
06:14 Example of CDF of a Discrete RV
13:53 Hybrid/Mixed (Continuous and Disrete) RV
18:32 Probability density function (pdf)
23:22 pdfs of continuous and discrete RVs
27:50 Probability mass associated with any segment of the real line using CDF/pdf
30:35 Examples of Continuous RVs
41:14 Examples of Discrete RVs
54:55 The binomial RV and the Normal (Gaussian) approximation

Lect 4 - Random Variables
1:37 Start
2:00 What is a random variable (r.v.)?
6:16 When you start defining events, there is no unified way to describe statistics across all types of experiments but mapping random variables onto the real line gives a common language to make sense out of the data.
7:12 Textbook/notes loose definition of r.v.
9:45 What is the function X in terms of r.v. ?
9:58 Drawing mapping events psi in the event space onto real line X points
12:18 Event subset A is getting mapped
14:01 If the set A which is (X^-1)*B also belings to the field, then it is an event and the probability of A is well-defined. R.v. must be mapped in a systematic way. Eq 3-1
15:25 if for every B on the real line you can find an A in the experiment space in the probability model, then you can associate a probability B on the real line.
16:42 the key thing is the notion of the r.v. and sigma field must have inverse mapping in order to determine any p(B) on the real line.
17:45 Getting a more specific definition of r.v. - a finite single-valued function of X that maps the set of all experimental outcomes into the set of real numbers is said to be a r.v. if this set is an event for every X in.
19:04 What is { ζ|X(ζ) ≤ x } in plain English?
20:06 Drawing explanation
22:08 The sample set can have elementary and complex events, and { ζ|X(ζ) ≤ x } refers to a specfic subset of events... It's a collection of events such that when you apply X, you land in a certain region B.
23:12 Re-reading specific definition of r.v. with new context
24:58 X is said to be r.v. if (X^-1)*B belongs to the sigma field F. All other sets can be constructed from these sets by performing set operations of unions, intersections, compliments, etc
26:00 x is r.v. and all psi's get mapped on real line but what about intervals {a ≤ X ≤ b} or {X=a}? Are those events?
27:32 if { ζ|X(ζ) ≤ x } holds then all other types of regions on the real line are valid events.
28:17 Complement of {X ≤ a} is an event, intersections are an event, etc.
28:51 Showing how a finite interval and all the intersections are also events. Eq 3-3. Drawing explanation. The intersection of all points in {a - 1/n < X ≤ a} is {X = a}
31:22 you start with experiment, you define a valid r.v. . You have probability on real line whether it's finite, semi-definite, or definite. Prob. of point depends on x
32:34 Example 3.2
35:16 Probability Distrbution Function (PDF) - characterizes the probability/statistical properties on the real line. (Note, this is different than pdf).
37:25 Probability P[( -inf, x )] is a fnct of x = Fx(x) (PDF/CDF)
39:27 Distribution Function definition - nondecreasing, right-continuous, and satisfies eq. 3-5
41:15 Shows that Fx(x) satisfies distribution function properties
41:36 (i) Fx(+inf) is the entire real line, so the whole set included, & P(set) = 1.
Fx(-inf) is nothing so it is the null set and P(null set) = 0
44:15 (ii) Consider x1 < x2, then (-inf, x1) is a subset of (-inf, x2) ...
45:22 Drawing explanation Fx(x1) < Fx(x2) which implies PDF is nonnegative and monotone nondecreasing
46:50 (iii) Right continuous nature. Consider x < xn-1 < ... < x2 < x1.
Consider event Ak = { ζ|x < X(ζ) ≤ xk } Eq 3-10. Drawing Explanation
50:26 The union { x < X(ζ) ≤ xk }∪{ X(ζ) ≤ x } = { X(ζ) ≤ xk} is semi-definite interval (-inf, xk) eq 3-11. The 2 regions are mutually exclusive so no overlap so you can add them.
51:05 P(Ak) = Fx(xk)-Fx(x) eq 3-12, Ak+1 is a subset Ak which is a subset of Ak-1 and so on. As K goes to infinity, the intersections of Ak's from 1 to inf becomes 0.
52:36 lim k -> inf (xk) is the right limit of xk (eq 3-14) so Fx(x) is right-continuous
53:45 (iv) If Fx(x0) = 0 for some x0, then Fx(x) = 0 for all x ≤ 0 eq(3-15). Also implies it is a null set
56:38 (v) The probability of X(ζ) > x is 1-Fx(x) eq 3-16
58:35 (vi) The probability of any finite interval x1 to x2 is the difference in the distribution functions Fx(x2)-Fx(x1), s.t. x2>x1 eq 3-17
1:00:10 (vii) The probability of a single point x can be expressed in terms of the distribution function as Fx(x) - its left limit Fx(x-) eq 3-18.
1:00:34 Consider x1= x - eps, s.t. and x2=x eps>0
You can use (vi) ---> lim eps-> 0 P(x - eps < X(ζ) ≤ x} = Fx(x) - lim eps---> 0 Fx(x - eps)
it is essentially Fx(x2) - Fx(x1) but substituted with eps which is squeezed infinitely small.
Eq 3-20, on the left this becomes P{X(ζ) = x} which equals Fx(x) - left limit Fx(x-) eq 3-20
1:01:39 If the function is continuous and Fx(x-) = Fx(x), what is prob of any single point x on the real line? Zero.
1:02:38 The probability is distributed on the real line that if you look a very narrow interval of a single point (no tolerances) that probability is 0 for discrete distributions. For continuous distributions, the probability isn't zero anymore
1:03:20 The right limit Fx(x0+) always = Fx(x), however the left limit Fx(x0-) may not equal fx(x0) and if that happens then the

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Fall 2020 Stochastic Processes Lecture 1