In the previous session (Part 9), we learned that the coefficient of $${\frac{1}{(1-t)^n}}$$ corresponds to combinations with repetition using the negative binomial theorem. In this session, we will ...

In the previous session (Part 1), using coin tosses as an example, we learned that the polynomial expansion rule of "choosing one term from each set of parentheses and multiplying them" acts as a ...

Probability underpins AI, cryptography and statistics. However, as the philosopher Bertrand Russell said, “Probability is the most important concept in modern science, especially as nobody has the ...

ABSTRACT: We obtain a closed form expression for the joint probability mass function of the occupation times for a Three-State Markov chain. Our representation extends the long-standing result for a ...

In many cases, we observe time series of discrete random variables with non-negative values, such as the number of customers or the number of insurance claims. The corresponding series of the first ...

remove-circle Internet Archive's in-browser bookreader "theater" requires JavaScript to be enabled. It appears your browser does not have it turned on. Please see ...

Life is uncertain. None of us know what is going to happen. We know little of what has happened in the past or is happening now outside our immediate experience. Uncertainty has been called the ...

As is well known, the normal distribution is a key tool in probability and statistics. It can be described as a distribution that obeys a universal rule derived from one of the most important theorems ...

When is it appropriate to completely reinvent the wheel? To an outsider, that seems to happen a lot in category theory, and probability theory isn’t spared from this treatment. We’ve had a useful ...

Once you have a basic understanding of sports betting odds, it’s easy to determine the favorite or underdog in a game and the favored option among a range of choices. You can also go a step further by ...