Probability, Chapter 5: Continuous Random Variables

Continuous random variable (5.1-5.3)

이상준 교수 (덕성여대 수학과) 2015년 2학기

Textbook: Sheldon Ross, A first course in probability (9th ed, Pearson)

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❖ Chapter 4: Discrete random variables

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Chapter 5: Continuous random variables

Contents❖ <5.1> Introduction

❖ <5.2> Expectation and Variance of Continuous Random Variables

❖ <5.3> Uniform Random Variable

❖ <5.4> Normal Random Variables

❖ <5.5> Exponential Random Variables

❖ <5.6> Other Continuous Distributions

❖ <5.7> Distribution of a Function of a Random Variable

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<5.1> Introduction❖ Example: There is a clock, and we roll the hour hand.

Let X be the time with the hour hand.

❖ Pr[0 ≤ X ≤ 12] =

❖ Pr[X=3] =

❖ Pr[X=5.8] =

❖ Pr[6≤X≤9] =

❖ Pr[7≤X≤12] =

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Probability density function →

Continuous random variable❖ � X is a continuous random variable if there is a nonnegative function f,

defined for all real x ∈ (-∞,∞), such that, for B ⊂ (-∞,∞),

❖ � f is called the probability density function of X.

(Reference: Ross, A first course in probability, 9th ed)

Property

(Reference: Ross, A first course in probability, 9th ed)

❖ Example 1a: Suppose that X is a continuous random variable whose probability density function is given by (a) What is the value of C?(b) Find P{X>1}.

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❖ Solution:

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❖ Example 1b: The amount of time in hours that a computer functions before breaking down is a continuous random variable with probability density function given by

❖ What is the probability that (a) a computer will function between 50 and 150 hours before breaking down? (b) it will function for fewer than 100 hours?

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❖ Solution:

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(Cumulative) Distribution fuction❖ Definition: The relationship between the cumulative distribution F and the

probability density f is expressed by

❖ Note: Differentiating both sides gives

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❖ Example 1d: If X is continuous with distribution function Fx and density function fx, find the density function of Y=2X.

❖ Solution:

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