1. Probability Theory Part 2: Random Variables

2. Random Variables  The Notion of a Random Variable The outcome is not always a number Assign a numerical value to the outcome of the experiment  Definition A function X which assigns a real number X(ζ) to each outcome ζ in the sample space of a random experiment S x SxSx ζ X(ζ) = x

3. Cumulative Distribution Function  Defined as the probability of the event {X≤x}  Properties x 2 1 F x (x) ¼ ½ ¾ 10 3 1 x

4. Types of Random Variables  Continuous Probability Density Function  Discrete Probability Mass Function

5. Probability Density Function  The pdf is computed from  Properties  For discrete r.v. dx f X (x) x

6. Conditional Distribution  The conditional distribution function of X given the event B  The conditional pdf is  The distribution function can be written as a weighted sum of conditional distribution functions where A i mutally exclusive and exhaustive events

7. Expected Value and Variance  The expected value or mean of X is  Properties  The variance of X is  The standard deviation of X is  Properties

8. More on Mean and Variance  Physical Meaning If pmf is a set of point masses, then the expected value μ is the center of mass, and the standard deviation σ is a measure of how far values of x are likely to depart from μ  Markov’s Inequality  Chebyshev’s Inequality  Both provide crude upper bounds for certain r.v.’s but might be useful when little is known for the r.v.

9. Joint Distributions  Joint Probability Mass Function of X, Y  Probability of event A  Marginal PMFs (events involving each rv in isolation)  Joint CMF of X, Y  Marginal CMFs

10. Conditional Probability and Expectation  The conditional CDF of Y given the event {X=x} is  The conditional PDF of Y given the event {X=x} is  The conditional expectation of Y given X=x is

11. Independence of two Random Variables  X and Y are independent if {X ≤ x} and {Y ≤ y} are independent for every combination of x, y  Conditional Probability of independent R.V.s