Calculate probabilities for linear combinations of independent normal random variables
Definition: Let \(X_{1}, X_{2}, \ldots, X_{n}\) be random variables and let \(c_{1}, c_{2}, \ldots, c_{n}\) be constants. Then, $$ \text{Y}=\text{c}_{1} \text{X}_{1}+\text{c}_{2} \text{X}_{2}+\cdots+\text{c}_{\text{n}} \text{X}_{\text{n}} $$ is a linear combination of \(X_{1}, X_{2}, \ldots, X_{n}\). In this reading, however, we will only…
Determine the distribution of order statistics from a set of independent random variables
Order Statistics Order Statistics are distributions obtained when we look at test scores from a random sample arranged in ascending order, i.e., from the smallest to the largest. In recent years, the importance of order statistics has increased because of…
Calculate joint moments, such as the covariance and the correlation coefficient for discrete random variables only
Let \(\text{X}\) and \(\text{Y}\) be two discrete random variables, with a joint probability mass function, \(\text{f}(\text{x}, \text{y})\). Then, the random variables \(\text{X}\) and \(\text{Y}\) are said to be independent if and only if, $$ \text{f}(\text{x}, \text{y})=\text{f}(\text{x}) * \text{f}(\text{y}), \quad \text…
Calculate variance and standard deviation for conditional and marginal probability distributions for discrete random variables only
Variance and Standard Deviation for Conditional Discrete Distributions Recall that, in the previous readings, we defined the conditional distribution function of \(X\), given that \(\text{Y}=\text{y}\) as: $$ \text{g}(\text{x} \mid \text{y})=\frac{\text{f}(\text{x}, \text{y})}{\text{f}_{\text{Y}}(\text{y})}, \quad \text { provided that } \text{f}_{\text{Y}}(\text{y})>0 $$ Similarly,…
Calculate moments for joint, conditional, and marginal discrete random variables
The \(n\)-th moment about the origin of a random variable is the expected value of its \(n\)-th power. In this reading, however, we will mostly look at moments about the mean, also called central moments. The \(n\)-th central moment of…
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
Joint Discrete Probability Distributions We are often interested in experiments that involve the intersection of two or more events. For example: An experimenter tossing a fair die is interested in the intersection of getting, say, a 5 and a 6….
Calculate the expected value, variance, and standard deviation of both the loss random variable and the corresponding payment random variable upon the application of policy adjustments.
In the previous reading, we covered the fundamental ideas of deductibles, coinsurance, benefit limits, and inflation in the context of insurance. In this reading, we will focus on more quantitative aspects. Specifically, we will perform calculations to determine the expected…
Apply the concepts of deductibles, coinsurance, benefit limits, and inflation to convert a given loss amount from a policyholder into the corresponding payment amount for an insurance company
Policy modifications refer to changes made to the loss random variable for an insurance product. In this chapter, we will explore several policy modifications, each serving a specific purpose in enhancing insurance coverage. These modifications include: Deductibles Benefit/policy limits Coinsurance…
Goals-based Approach
A goals-based asset allocation process combines into an overall portfolio numerous sub-portfolios, each designed to fund a single goal within its time horizon and required probability of success. Individuals have unique needs that differ from those of institutions. The critical…
