02:37

All video materials are available on ptfuture.net now! Check it out guys!

20:57

std::atomic memory orders. Compare relaxed, consume, acquire, release, sequence consistent mem order

13:49

Conditional Expectation Mean square error and Orthogonality, 𝐸[𝑋𝑌|F]=𝑋𝐸[𝑌|F] | Martingale Theory

18:16

Deriving the conditional expectation from joint probability density function | Martingale Theory

14:44

Proof: Bayes’ Formula and Conditional Probability formula | Martingale Theory

00:36

Bitcoin SV 51% attack | Random Walk Markov Chain process modelling

18:22

Conditional Expectation and Radon-Nikodym Theorem | Martingale Theory

13:02

C++ 11 to 20: Boost Coroutine2. What Why and How. Learn Boost Coroutine async programming by example

03:16

C++ 11 to 20: Dead lock prevention using std::lock

15:22

Sufficient and necessary conditions for Almost sure convergence

03:22

Prove: Integrability implies finite almost surely

12:45

C++ 11 to 20, Comparing different function types: Function pointer, std::function, functor, lambda

10:46

Prove Kolmogorov's 0-1 law: If 𝑋1,𝑋2,... are independent and 𝐴∈Tail σ field then 𝑃(𝐴)=0 or 𝑃(𝐴)=1.

10:41

All pairwise correlations equal: corr(X1,X2) = corr(X1,X3) = corr(X2,X3) = ρ. What's the range of ρ?

22:12

Definition of Tail 𝜎-field and Examples, lim S_n, limsup S_n, lim S_n/n etc.

03:39

Monte Carlo to the rescue! Solving the "Expected sum of distances in a triangle" Question in C++!

10:07

Brain teasers! - Expected sum of distances in a triangle

17:49

Prove Central Limit Theorem using characteristic function

13:15

Prove: 𝜙𝑛(𝑡)→𝜙(𝑡) and 𝜙(𝑡) is continuous at 0 indicates 𝐹𝑛 is tight & 𝐹𝑛⇒𝐹 where F is a distribution

03:32

Theorem: Sufficient condition for tightness (Bounded in probability)

06:40

Every subseq’s limit func 𝐹 in Helly’s selection theorem is a distribution function iff 𝐹𝑛 is tight

05:56

Prove Helly’s selection theorem

03:31

Definition: Tightness or Bounded in Probability

01:31

Prove If 𝑋𝑛⇒𝑋∞ then 𝜙𝑛(𝑡)→𝜙(𝑡) for all 𝑡.

08:39

Prove Slutsky’s theorem. If 𝑋𝑛⇒𝑋, 𝑌𝑛→𝑐 in prob, 𝑍𝑛→𝑑 in prob, 𝑍𝑛+𝑌𝑛𝑋𝑛⇒𝑑+𝑐𝑋. If 𝑐≠0, 𝑍𝑛+𝑋𝑛/𝑌𝑛⇒𝑑+𝑋/𝑐.

05:06

𝑔 be a measurable function with 0 measure discont pts, 𝑔(𝑋𝑛)⇒𝑔(𝑋∞). If 𝑔 is bounded, 𝐸𝑔(𝑋𝑛)→𝐸𝑔(𝑋∞).

05:41

If 𝑋𝑛⇒𝑋 and 𝑌𝑛−𝑋𝑛→0 in prob, then 𝑌𝑛⇒𝑋 in dist. If 𝑋𝑛⇒𝑋, 𝑍𝑛→0 in prob, then 𝑋𝑛+𝑍𝑛⇒𝑋 in dist.

13:56

Proof: Portmanteau Lemmas

07:04

Proof 𝑋𝑛⇒𝑏 shows 𝑋𝑛→𝑏 in prob. 𝑋𝑛→𝑋 in prob shows 𝑋𝑛⇒𝑋. But 𝑋𝑛⇒𝑋 not indicate 𝑋𝑛→𝑋 in general.

02:55

Proof: 𝑋𝑛+𝑌𝑛→𝑋+𝑌 in probability, 𝑋𝑛𝑌𝑛→𝑋𝑌 in probability

01:44

Proof: Continuous function preserves convergence in probability.

06:13

Prove: 𝑋𝑛→𝑋 in probability iff for every subseq 𝑋𝑛(𝑚) there’s a further subseq𝑋𝑛(𝑚𝑘)→𝑋 almost surely

13:22

Machine Earning & Investment | Pricing and arbitrage in Forward & Future Contracts

09:40

R language Data Types, Data Frame, List and Factor

39:07

Characteristic function | Fourier Transform | Inversion Theorem

38:46

Proof of Strong law of large numbers and weak law of large numbers

03:27

Finding you

27:17

R-lang Vector and Matrix Types | R Language Intro for Data Scientists and Statisticians

03:09

Flipping coins | Prove observing all tails in an experiment has probability 0 using Borel Cantelli

11:00

Proof of Borel Cantelli Lemmas | limsup liminf | infinitely often and eventually all events

16:02

Jensen’s, Holder’s, Chebyshev’s inequality | Fatou’s lemma | Dominiated convergence | Fubini Theorem

02:19

Holding On To You

31:42

Probability Measure | Sigma Field, Borel Sets, Lebesgue Measure, Definition of Random Variable

23:56

Introduction to Hidden Markov Chain and Viterbi Algorithm

07:13

Music: You are unbreakable

09:26

What is time reversible Markov Chain | Theory and Proof

06:29

Limiting Stationary Distribution of Markov Chain | Asymptotic Behaviors

06:03

We'll get there soon... We'll get there soon... We'll get there soon...

05:59

Markov Chain Monte Carlo | Gibbs Sampler Algorithm

14:35

Markov Chain Monte Carlo | Metropolis–Hastings Algorithm | Theory Explained

04:53

Statistical Test of Independence for two normal samples | Sample Correlation Coefficient Method

06:17

Jumping to the thousands - 0002

40:28

Get started with Simple Linear Regression with One Predictor variable | Not for Beginners Guide

08:17

Least Squared Method and Projection Matrix | Regression Analysis

17:01

How effective are the vaccines? Quadratic Forms and Statistical Inferences on Normal models

01:47

ptfuture.net Website Launched | Predicting the Future Mathematically

21:34

Uniqueness of Stationary Distribution | Markov Chain Theory Episode 4

25:15

Markov Chain Theory Episode 3 | Under what condition does stationary distribution exist?

46:29

What is null recurrent? What is positive recurrent? Must Random variable's Probabilities sum to one?

11:17

Bitcoin and Ethereum are not as safe as you think | Markov chain in Blockchain