Lesson 3.1 PDF, Mean, Var, CDF

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McGuffey Spring 2023 Stat 310 Notes L ESSON 3.1: C ONTINUOUS RV S : CDF, PDF, M EAN & V ARIANCE 1. Cumulative Distribution Functions A cumulative distribution function (CDF) is another way to describe the distribution of a random variable. Its definition is the same for discrete and continuous random variables. C UMULATIVE D ISTRIBUTION F UNCTIONS The cumulative distribution function (CDF) of an r.v. X is the function F X given by Example 1. Consider X with PMF x 2 4 6 8 P ( X = x ) 0.4 0.2 0.1 0.3 . (a) Plot the PMF of X . (b) Describe the CDF of X with a table. That is, find P ( X x ) for x = 2, 4, 6, 8. This is an incomplete way to describe the CDF, but it helps us see what's going on. (c) Plot the CDF of X . Lesson 3.1 Page 1/5
McGuffey Spring 2023 Stat 310 Notes CDF Visualizations For Named Distributions • Discrete and continuous: https://www.geogebra.org/m/cQeyMaaO ! 4 Not our version of the Geometric distribution. Example 2. Consider a new r.v. X , with CDF plotted below. Provide the PMF of X in table form. CDF Properties A valid CDF F X has the following properties: 1. : If x 1 x 2 , then F ( x 1 ) F ( x 2 ). 2. Tends to as x → -∞ . 3. Tends to as x → ∞ . 2. Continuous Random Variables Formal definition: A random variable X is called continuous if there is a nonnegative function f X called the probability density function (PDF) of X , such that P ( X B ) = Z B f X ( x ) dx , for every subset B of the real line. Lesson 3.1 Page 2/5
McGuffey Spring 2023 Stat 310 Notes Discrete RVs Continuous RVs Possible Values distinct values Distributions PMF: p X ( x ) = P ( X = x ) PDF: f X ( x ) CDF: F X ( x ) = P ( X x ) CDF: F X ( x ) = P ( X x ) Valid PMF/PDF Non-negative. p X ( x ) 0 for any x R Total mass of 1. x p X ( x ) = 1 Non-negative. f X ( x ) 0 for any x R Total area of 1. Discrete RVs Continuous RVs Probability Calculations P ( a X b ) = P ( a X b ) = X a x b p X ( x ) P ( X = b ) = p X ( b ) P ( X = b ) = P ( X b ) = x b p X ( x ) P ( X b ) = = F X ( b ) Including Endpoints ! 4 It matters. On problem sets (but not exams), you may use WolframAlpha to evaluate integrals and take derivatives. On exams, you may use an approved calculator. Always, you must include in your solution a statement of the integral or derivative being evaluated. Lesson 3.1 Page 3/5
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