Covariance between variables can be calculated in two ways. One method is the historical sample covariance between two random variables \(X_i\) and \(Y_i\). It is based on a sample of past data of size \(n\) and is given by: $$\text{Cov}_{X_i,Y_i}=\frac{\sum_{i=1}^{n}{(X_i -\bar{X})(Y_i…

A tree diagram is a visual representation of all possible future outcomes and the associated probabilities of a random variable. Tree diagrams are particularly useful when we have several possible outcomes. They facilitate the recording of all the possibilities in…

In the context of investments, conditional expectation refers to the expected value of an investment, given a certain set of real-world events that are relevant to that particular investment. This means that in their calculation and prediction of the expected…

Expected Value The expected value of a random variable is the average of the possible outcomes of that variable, taking the probability weights into account. Therefore: $$ E\left( X \right) =\sum _{ i=1 }^{ n }{ { X }_{ i…

We can use the total probability rule to determine the unconditional probability of an event in terms of conditional probabilities in certain scenarios.

Two or more events are independent if the occurrence of one event does not influence the occurrence of the other event(s). Let us put this in annotations:

A cumulative distribution function, \(F(x)\), gives the probability that the random variable \(X\) is less than or equal to \(x\): $$ P(X ≤ x) $$ By analogy, this concept is very similar to the cumulative relative frequency.

Unconditional Probability Unconditional probability (also known as marginal probability) is simply the probability that the occurrence of an event does not, in any way, depend on any other preceding events. In other words, unconditional probabilities are not conditioned on the…

Defining properties of a probability refers to the rules that constitute any given probability. These are:

Odds for and against an event represent a ratio of the desired outcomes versus the field. In other words, the odds for an event are the ratio of the number of ways the event can occur to the number of…