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Chapter 5. Discrete Random Variables and Probability Distributions

Unit 5.18. Questions for repetition

1. List and prove all properties of means.
2. List and prove all properties of covariances.
3. List and prove all properties of variances. How would you show that variance does not have the linearity property?
4. List and prove all properties of standard deviations. Can you prove that $latex \sigma (X + Y) \le \sigma (X) + \sigma (Y)$?
5. List and prove all properties of the correlation coefficient including the statistical interpretation of different cases.
6. How and why do you standardize a variable?
7. Define and derive the properties of the Bernoulli variable.
8. Explain the term "independent identically distributed".
9. Define a binomial distribution and derive its mean and variance with explanations.
10. Prove equation Eq. (5.15) (see Distribution of the binomial variable).
11. Define a distribution function, describe its geometric behavior and prove the interval formula.
12. Is the sum of two distribution functions a distribution function? Hint: does $latex {F_X}(t) + {F_Y}(t)$ satisfy properties from Distribution function?
13. Is the product of two distribution functions a distribution function?
14. Is $latex F(x) = \frac{1}{{1 + {e^{ - x}}}}$ a distribution function? What is its derivative $latex f(x)$? What is the link between $latex F(x),f(x)$?
15. Define the Poisson distribution and describe (without proof) how it is applied to the binomial distribution.
16. Calculate the mean and variance of the Poisson distribution.
17. Prove that the rate of return on the portfolio is a linear combination of the rates of return on separate assets.
18. (Challenging) By definition, the distribution function of two random variables $latex {F_{X,Y}}(x,y)$ is the probability of the joint event $latex {X \le x,Y \le y}$. In this case, what would be the analog of the properties from Distribution function? How does Eq. (5.16) generalize for this case?

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