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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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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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Commodity Life Cycles

The life cycle of commodities differs significantly depending on the economic, technical, and structural (i.e., industry, value chain) profile of each commodity and the sector. The commodity life cycle has an impact on the following aspects: It reflects and magnifies…

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FRM Questions Images

FRM 1 MOCK 1 Reference tables Question 84 Question 86 FRM 1 MOCK 2 Question 71 Question 72 FRM 2 MOCK 1 FRM 2 MOCK 2

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Stress-Testing

After completing this reading, you should be able to: Describe the rationale for the use of stress testing as a risk management tool. Identify key aspects of stress testing governance, including choice of scenarios, regulatory specifications, model building, stress-testing coverage,…

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