Transformation for Bivariate Discrete Random Variables Let \(X_1\) and \(X_2\) be a discrete random variables with joint probability mass function \(f_{X_1,X_2}(x_1,x_2)\) defined on a two dimensional set \(A\). Define the following functions: $$ y_1 =g_1 (x_1, x_2)$$ and  $$y_2  =g_2(x_1,x_2)$$…

Let \(X\) and \(Y\) be two discrete random variables, with a joint probability mass function, \(f\left(x, y\right)\). Then, the random variables \(X\) and \(Y\) are said to be independent if and only if, $$ f\left(x,\ y\right)=f\left(x\right)\times f\left(y\right),\ \ \ \…

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…

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…

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…

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…

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…