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General Cash Flows, Portfolios, and Asset Liability Management

After completing this chapter, the candidate will be able to: Define and recognize the definitions of the following terms: yield rate/rate of return, current value, duration and convexity (Macaulay and modified), portfolio, spot rate, forward rate, yield curve, cash flow…

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Apply the Central Limit Theorem to calculate probabilities for linear combinations of independent and identically distributed random variables

One of the most important results in probability theory is the central limit theorem. According to the theory, the sum of a large number of independent random variables has an approximately normal distribution. The theory provides a simple method of…

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Calculate moments for linear combinations of independent random variables

In the previous reading, we defined \(Y=c_{1} X_{1}+c_{2} X_{2}+\cdots+c_{p} X_{p}\) to be a linear combination of the independent random variables \(X_{1}, X_{2}, \ldots, X_{p}\) where \(c_{1}, c_{2}, \ldots, c_{p}\) are constants. Now, we may wish to calculate moments such as…

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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…

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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…

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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…

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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,…

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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…

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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…

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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….

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