Probability Theory Part 2: Random Variables
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
Cumulative Distribution Function Defined as the probability of the event {X≤x} Properties x 2 1 F x (x) ¼ ½ ¾ x
Types of Random Variables Continuous Probability Density Function Discrete Probability Mass Function
Probability Density Function The pdf is computed from Properties For discrete r.v. dx f X (x) x
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
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
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.
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
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
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