Probability Theory

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 ζ X(ζ) = x x Sx

3. Cumulative Distribution Function • Defined as the probability of the event {X≤x} • Properties Fx(x) 1 x Fx(x) 1 ¾ ½ ¼ 3 x 0 1 2

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

5. Probability Density Function • The pdf is computed from • Properties • For discrete r.v. fX(x) fX(x) dx x

6. 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 Ai mutally exclusive and exhaustive events Conditional Distribution

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

8. 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. More on Mean and Variance

9. 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 Joint Distributions

10. 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 Conditional Probability and Expectation

11. 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 Independence of two Random Variables