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…
Exam P Images
Chapter 1-General Probability Reading 1 Question 4
Exam P Syllabus – Learning Outcomes
General Probability 1.a – Define set functions, Venn diagrams, sample space, and events. Define probability as a set function on a collection of events and state the basic axioms of probability. 1.b – Calculate probabilities using addition and multiplication rules. 1.c – Define…
State and apply the Central Limit Theorem
For this learning objective, a certain knowledge of the normal distribution and knowing how to use the Z-table is assumed. The central limit theorem is of the most important results in the probability theory. It states that the sum of…
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, $$ Y=c_1X_1+c_2X_2+\ldots+c_nX_n $$ is a linear combination of \(X_1, X_2,\ldots, X_n\). In this reading, however, we will only base our discussion on the linear combinations…
Determine the distribution of a transformation of jointly distributed random variables
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 $$y_2 =g_2(x_1,x_2)$$…
Calculate joint moments, such as the covariance and the correlation coefficient
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),\ \ \ \…
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…
