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An overview of mass functions and density functions for discrete and continuous random variables. It covers the definition, properties, and calculations of mass functions for discrete random variables, and the relationship between mass functions and density functions for continuous random variables. The document also includes exercises to help students understand the concepts.

Typology: Study Guides, Projects, Research

2021/2022

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Download Probability Distributions for Discrete and Continuous Variables and more Study Guides, Projects, Research Calculus in PDF only on Docsity! Mass Functions Density Functions Topic 7 Random Variables and Distribution Functions Mass Functions and Density Functions 1 / 9 Mass Functions Density Functions Outline Mass Functions Density Functions 2 / 9 Mass Functions Density Functions Mass Functions Recall that a geometric sequence c , cr , cr2, . . . , crn has sum sn = c + cr + cr2 + · · ·+ crn = c(1− rn+1) 1− r for r 6= 1. If |r | < 1, then limn→∞ rn = 0 and thus sn has a limit as n→∞. In this case, the infinite sum is the limit c + cr + cr2 + · · ·+ crn + · · · = lim n→∞ sn = c 1− r . Thus, ∑∞ x=0 fX (x) = ∑∞ x=0(1− p)xp = p 1−(1−p) = 1. F (b) = P{X > b} = ∞∑ x=b+1 fX (x) = ∞∑ x=b+1 (1− p)xp = (1− p)b+1p 1− (1− p) = (1− p)b+1 and FX (b) = 1− (1− p)b+1 for b = 0, 1, 2, . . .. 5 / 9 Mass Functions Density Functions Mass Functions Exercise. We use R to investigate a geometric random variable with p = 1/4. Enter the commands > x<-c(0:10) #creates a sequence from 0 to 10 > f<-dgeom(x,1/4) #gives the mass function for these values > F<-pgeom(x,1/4) #gives the distribution function for these values > data.frame(x,f,F) • Check that the jumps in the cumulative distribution function FX (x)− FX (x − 1) is equal to the values of the mass function. • Find 1. P{X ≤ 4}, 2. P{2 < X ≤ 5}, and 3. P{X ≥ 5}. 6 / 9 Mass Functions Density Functions Density Functions For X a random variable whose distribution function FX has a derivative. The function fX satisfying FX (x) = ∫ x −∞ fX (t) dt is called the probability density function and X is called a continuous random variable. By the fundamental theorem of calculus, the density function is the derivative of the distribution function. fX (x) = lim ∆x→0 FX (x + ∆x)− FX (x) ∆x = F ′X (x). In other words, if ∆x is small, FX (x + ∆x)− FX (x) ≈ fX (x)∆x . 7 / 9