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Introduction The fourth post of my probability series is about  probability distributions . Basic Probability Two Random Variables Chain Rule of Probability Theory In the previous posts, I discussed probabilities involving a single and multiple random variables. Briefly, a random variable in an experiment or a trial maps a specific outcome from a sample space to a numeric value.  The example of a probabilistic event which I kept using was whether I to go to Paris next year \(X=1\) or not \(X=0\). A probability of a specific event can be expressed like this: \[ P(X=1) = 0.7 \] The generalised probability is then like this: \[ P(X) \] where \(X\) can be anything in a defined sample space. A probability distribution concerns probabilities of all possible outcomes of a sample space. So, \( P(X=1)=0.7 \) is a part of a probability distribution, but the whole representation of the probability distribution is \( P(X) \). More concrete examples are: we think about "going to Paris...

Introduction  This is my third post on the probability theory: Basic Probability Two Random Variables The main focus of the series is on easy introduction to probability theory. A bit of recap : in the last post of Two Random Variables, I discussed probabilities of two random events including joint, conditional and marginal probabilities. That post was about probabilities of two dependent and independent events. The example was whether Emily and I go to Paris next year. If Emily and I did not know each other, Emily visiting Paris next year would have no impact on my chance of going to Paris (two events were independent). On the other hand, if I knew Emily and she text me about her trip to Paris, that could affect my chance of visiting Paris next year (two events were dependent). This post focuses on generalisation of multiple events and random variables. So, I am introducing a new event. An alien is visiting Paris. We need to think about how this event will impact chances of my and...

Introduction In the previous post of  Basic Probability , I discussed my chance of visiting Paris next year. I had a sample space like this: \( S = \{Meet, No\_more\_holiday, No\_money, Paris\_gone\_from\_Earth, ...\} \). The random variable of \( X \) was all about me going to Paris next year. What I am going to write in this post is when we have two sample spaces, two outcomes and two random variables. Having two random variables means that we need to consider the followings: two events happen simultaneously ( joint probability ), only one event happens regardless of the other event ( marginal probability ) and one event happens because of the other event ( conditional probability ). Let's define a simple sample space of me visiting Paris next year is \( S_X = \{me\_in\_paris, me\_not\_in\_paris\} \).  Let's define another sample space to have two events at the same time. The second event is whether Emily goes to Paris next year. We now have anot...