Raising the bar

Chapter 6. Continuous Random Variables and Probability Distributions

Unit 6.10. Questions for repetition

1. Discuss the five properties of integrals that follow their definition: do summations have similar properties?

2. How do you prove that a density is everywhere nonnegative?

3. How do you justify the definition of the mean of a continuous random variable?
4. With the knowledge you have, which of the 18 properties listed in Mean, covariance, variance and correlation can you prove? Mark with a star those you can't prove at this point. Some additional definitions and proofs will be given later.
5. Define the uniformly distributed continuous random variable. Find its mean, variance and distribution function.
6. In one block give the properties of the standard normal distribution, with proofs, where possible, and derive from them the properties of normal variables.
7. What is the relationship between distribution functions $latex {F_z},\,\,{F_X}$ if $latex X = \sigma z + \mu$? How do you interpret $latex {F_z}(1) - {F_z}(-1)$?
8. Divide the statement of the central limit theorem into a preamble and convergence statement. Provide as much justification/intuition for the preamble as you can. How is this theorem applied to the binomial variable?
9. Draw a parallel between the proofs of multiplicativity of means for discrete and continuous random variables.

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