Calculate variance, standard deviation for conditional and marginal probability distributions
Variance and Standard Deviation for Conditional Discrete Distributions In the previous readings, we introduced the concept of conditional distribution functions for random variable \(X\) given \(Y=y\) and the conditional distribution of \(Y\) given \(X=x\). We defined the conditional distribution function…
Explain and apply joint moment generating functions
We can derive moments of most distributions by evaluating probability functions by integrating or summing values as necessary. However, moment generating functions present a relatively simpler approach to obtaining moments. Univariate Random Variables In the univariate case, the moment generating…
Calculate moments for joint, conditional, and marginal random variables
Moments of a Probability Mass function The n-th moment about the origin of a random variable is the expected value of its n-th power. Moments about the origin are \(E(X),E({ X }^{ 2 }),E({ X }^{ 3 }),E({ X }^{ 4 }),….\quad\) For…
Determine conditional and marginal probability functions for discrete random variables only
Marginal Probability Distribution In the previous reading, we looked at joint discrete distribution functions. In this reading, we will determine conditional and marginal probability functions from joint discrete probability functions. Suppose that we know the joint probability distribution of two…
Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only
Discrete Joint Probability Distributions In the field of probability and statistics, we often encounter experiments that involve multiple events occurring simultaneously. For example: An experimenter tossing a fair die is interested in the intersection of getting, say, a 5 and…
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)}…
