Apply Transformations
Transformations allow us to find the distribution of a function of random variables. There are different methods of applying transformations. The Method of Distribution Function Given a random variable \(Y\) that is a function of a random variable \(X\), that…
Determine the sum of independent random variables (Poisson and normal)
The Sum of Independent Random Variables Given \(X\) and \(Y\) are independent random variables, then the probability density function of \(a=X+Y\) can be shown by the equation below: $$ { f }_{ X+Y }\left( a \right) =\int _{ -\infty }^{…
Probability Generating Functions and Moment Generating Functions
Probability Generating Function The probability generating function of a discrete random variable is a power series representation of the random variable’s probability density function as shown in the formula below: $$ \begin{align*} \text{G}\left(\text{n}\right)&=\text{P}\ \left(\text{X}\ =\ 0\right)\bullet \ \text{n}^0\ +\ \text{P}\…
Explain and calculate variance, standard deviation, and coefficient of variation
Variance The mean (average) gives an idea of the “typical” value in a dataset. However, in many scenarios, especially in financial markets, simply knowing the mean isn’t sufficient. Investors want to know not just the expected return (mean) but also…
Explain and calculate expected value and higher moments, mode, median, and percentile
Expected Value of Discrete Random Variables Let \(X\) be a discrete random variable with probability mass function, \(p(x)\). The expected value or the mean of the random variable \(X\), denoted as \(E(X)\), is given by: $$ E \left(X\right)=\sum{x. p (x)}…
Explain and apply the concepts of random variables
Definitions: Variable: In statistics, a variable is a characteristic, number, or quantity that can be measured or counted. Random variable: A random variable (RV) is a variable that can take on different values, each with a certain probability. It essentially…
State Bayes Theorem and use it to calculate conditional probabilities
Bayes Theorem Before we move on to Bayes Theorem, we need to learn about the law of total probability. The Law of Total Probability The law of total probability states that if E is an event, and \(A_1, A_2, \cdots A_n\)…
Calculate probabilities using combinatorics, such as combinations and permutations
The Multiplication Principle of Counting Assume that you are conducting an experiment where the outcomes consist of combining two separate actions or tasks. As such, assume that there are \(n\) possibilities for the first task and that for each of…
Define and calculate conditional probabilities
Conditional Probability Given two events \(A\) and \(B\), the conditional probability of event \(A\) occurring, given that event \(B\) has occurred, is the probability of event \(A\) and \(B\) occurring over the probability of event \(B\) occurring as shown by…
Define independence and calculate probability of independent events
Define independence and calculate probabilities of independent events 1.1 Defining Independence In probability and statistics, two events, \(A\) and \(B\), are said to be independent if the occurrence of event A does not affect the probability that the other event…
