MA : Introductory Probability
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1 MA : Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky Spring 2017 David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
2 Cumulative Distribution Functions Definition Let X be a continuous real-valued random variable. Then the cumulative distribution function or c.d.f. of X is defined by the equation F(x) = P(X x) David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
3 Cumulative Distribution Functions Definition Let X be a continuous real-valued random variable. Then the cumulative distribution function or c.d.f. of X is defined by the equation F(x) = P(X x) Here F(x) accumulates (or, more simply, cumulates) all of the probability less than or equal to x. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
4 Cumulative Distribution Functions The density function and the cumulative distribution function are related in the following way. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
5 Cumulative Distribution Functions The density function and the cumulative distribution function are related in the following way. Theorem Let X be a continuous real-valued random variable with density function f (x). Then the function defined by F(x) = x f (t)dt, < x < is the cumulative distribution function of X. Furthermore, by the FTC, we have F (x) = f (x). David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
6 Example A real number is chosen at random from [0, 1] with uniform probability, and then this number is squared. Let X represent the result. 1 What is the c.d.f. of X? 2 What is the density function of X? David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
7 Example A real number is chosen at random from [0, 1] with uniform probability, and then this number is squared. Let X represent the result. 1 What is the c.d.f. of X? 2 What is the density function of X? Solution 0 if, x 0 F(x) = x if, 0 x 1 1 if, x 1. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
8 Example A real number is chosen at random from [0, 1] with uniform probability, and then this number is squared. Let X represent the result. 1 What is the c.d.f. of X? 2 What is the density function of X? Solution 0 if, x 0 F(x) = x if, 0 x 1 1 if, x 1. 0 if, x 0 1 f (x) = 2 if, 0 x 1 x 0 if, x 1. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
9 Consider a random variable defined to be the sum of two random real numbers chosen uniformly from [0, 1]. Let the random variables X and Y denote the two chosen real numbers. Define Z = X + Y. 1 What is the c.d.f. of Z? 2 What is the density function of Z? David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
10 Consider a random variable defined to be the sum of two random real numbers chosen uniformly from [0, 1]. Let the random variables X and Y denote the two chosen real numbers. Define Z = X + Y. 1 What is the c.d.f. of Z? 2 What is the density function of Z? Solution 0 if, z 0 z 2 2 if, 0 z 1 F Z (z) = 1 (2 z)2 2 if, 1 z 2 1 if, z 2. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
11 Consider a random variable defined to be the sum of two random real numbers chosen uniformly from [0, 1]. Let the random variables X and Y denote the two chosen real numbers. Define Z = X + Y. 1 What is the c.d.f. of Z? 2 What is the density function of Z? Solution 0 if, z 0 z 2 2 if, 0 z 1 F Z (z) = 1 (2 z)2 2 if, 1 z 2 1 if, z 2. 0 if, z 0 z if, 0 z 1 f (z) = 2 z if, 1 z 2 0 if, z 2. David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
12 Example (Homework 2 Problem 9) Take a stick and break it into two pieces, choosing the break point at random from a uniform distribution. What is the probability that the larger piece is no more than 2.8 times as long as the smaller piece? Solution David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
13 Example (Homework 2 Problem 9) Take a stick and break it into two pieces, choosing the break point at random from a uniform distribution. What is the probability that the larger piece is no more than 2.8 times as long as the smaller piece? Solution ( ) ( ) David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
14 Example (Homework 2 Problem 12) Suppose you choose a real number X from the interval [a,b] with the density function f (x) = Cx, where C is a constant. a) Find C. b) Find P(E), where E = [a, b] is a subinterval of [a,b] (as a function of a and b). c) Find P(X > 8). d) Find P(X < 13). e) Find P(X 2 21X ). Solution David Murrugarra (University of Kentucky) MA 320: Section 2.2 Spring / 7
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