Probability distributions are mathematical representations of the probability of different outcomes in an experiment or event.
A discrete probability distribution is a probability distribution that deals with a countable number of outcomes. In other words, it is a distribution that deals with variables that can only take on certain values. These variables can only take on values that are distinct and separate from each other. For example, the outcome of rolling a die is discrete because there are only six possible outcomes: 1, 2, 3, 4, 5 or 6.
On the other hand, a continuous probability distribution is a probability distribution that deals with variables that can take on any value within a given interval. This means that the outcomes are not discrete and can include all possible values within the interval. Continuous distributions are often associated with measurements and physical phenomena. For example, the height of a person is a continuous distribution because it can take on any value within a given range.
In terms of quantitative trading/investment, probability theory is applied to model the behavior of financial instruments such as stocks, options, and futures. Discrete probability distributions are often used to model the behavior of discrete financial events such as stock prices that only change in small increments. For example, a binomial distribution can be used to model the price change of a stock that can either go up or down by a certain amount.
Continuous probability distributions, on the other hand, are often used to model the behavior of continuous financial events such as interest rates or returns. For example, a normal distribution can be used to model the returns of a portfolio of stocks over time.