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7/23/2014 What is a Random Variable? Online Review Course of Undergraduate Probability and Statistics • Random Variable: a real-valued function of the outcomes of the experiment Review Lecture 9 Continuous Random Variables – Ω→ (maps sample space onto the real numbers) • Example – Ω = {all UT students}, X = height of randomly selected student Chris A. Mack • Discrete versus Continuous random variable Adjunct Associate Professor – Option 1: round height measurement to the nearest inch. Result = discrete RV – Option 2: measure height with infinite precision. Result = continuous RV. Course Website: www.lithoguru.com/scientist/statistics/review.html © Chris Mack, 2014 1 © Chris Mack, 2014 Continuous Random Variable 2 Probability Density Function • Consider a discrete RV: • Definition of PDF: lim • Now let the distance between values of x go to PDF, probability density function zero • Probability is the area under the pdf curve Replace summation with integral Replace PMF(x) with PDF(x) dx a © Chris Mack, 2014 3 • Normalization: 4 Expectation and Variance 0 Discrete: "# $ 1 0 • The probability at a point is zero: • Example: Uniform probability a b % &# $ ! "# $ ' Continuous: 1 ! © Chris Mack, 2014 b © Chris Mack, 2014 PDF Properties • Non-negativity: Probability per unit length along x → " 5 © Chris Mack, 2014 % & ! "# $ ' 6 1 7/23/2014 Uniform PDF Cumulative Distribution Function • Uniform pdf between a and b, zero outside this range ( " 1/* ! + 0 ! % & 5 ,-./&012/ ' 2* ! + ! "# $ • CDF for a continuous RV 4 • CDF is monotonically non-decreasing 5 b ' © Chris Mack, 2014 2 ! 12 5 ' 7 © Chris Mack, 2014 Normal Distribution 5 * + 267 / 1 1 2 " © Chris Mack, 2014 % & 8 9 ':9 /& < • Define > B ;*<, 7 ' + !< 9 ; 0,1 ; @< A, @7 ' © Chris Mack, 2014 10 • How are discrete and continuous RVs related? • How are PDF and CDF related for a continuous RV? • Know how to calculate expectation and variance given a PDF. • What is the normal distribution and some of its important properties? Histograms are used to compare to PDFs • Wikipedia is a very good source for distributions and their properties © Chris Mack, 2014 ; <, 7 ' -./CD Review #9: What have we learned? • Some common PDFs: Exponential distribution Cauchy distribution Student’s t-distribution Chi-squared distribution Log-normal distribution Beta distribution Gamma distribution Etc. “standard normal” (z-score) – Let ? @ A. – If ; <, 7 ' then ? There are Many, Many PDFs – – – – – – – – 8 : • A linear function of a normally distributed RV produces a normally distributed RV 27 7' 8 Normal Distribution • Also called the Gaussian distribution 1 !5 11 © Chris Mack, 2014 12 2