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lecture8 - Stat 302, Introduction to Probability Jiahua...

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Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 8 January-April 2011 1 / 24
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Example: Uniform distribution In many applications, the random quantity is equally likely over some closed interval. The corresponding random variable should hence has constant density function over this interval. The simplest one is f ( x ) = b 1, 0 x 1; 0, otherwise. For simplicity, we may just write the portion: f ( x ) = 1 for 0 < x < 1. Jiahua Chen () Lecture 8 January-April 2011 2 / 24
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Example: Uniform distribution For a uniformly distributed random variable on [ 0, 1 ] as we just introduced, F ( x ) = 0, x < 0; x , 0 x 1; 1, x > 1. When we discuss continuous random variables, it is not as essential to specify the Random experiment, Sample space, Events and probability measure . When we get confused, these notions are helpful to have certain issues clari±ed. Jiahua Chen () Lecture 8 January-April 2011 3 / 24
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Example: Exponential distribution Let X be a uniformly distributed random variable on [ 0, 1 ] as we just discussed. Let Y = - λ - 1 ln ( X ) where λ > 0 is a non-random constant. By deFnition, X is a function on some sample space Ω , so Y is also a function on Ω , albeit a composite function. Consequently, Y is a random variable. Jiahua Chen () Lecture 8 January-April 2011 4 / 24
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Example: Exponential distribution It is seen that { ω : Y ( ω ) t } = { ω : X ( ω ) > exp ( - λ t ) } . Hence, for any t > 0, we have P ( Y t ) = P ( X > exp ( - λ t )) = 1 - exp ( - λ t ) , and when t < 0 we have P ( Y t ) = 0. Jiahua Chen () Lecture 8 January-April 2011 5 / 24
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Example: Exponential distribution In other words, Y has cdf F y ( y ) = b 0, y < 0; 1 - exp ( - λ y ) , y > 0. and its pdf is on y > 0 with f y ( y ) = λ exp ( - λ y ) . Because its density function is an exponential function, it is named as Exponential distribution . Jiahua Chen () Lecture 8 January-April 2011 6 / 24
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Normally distributed random variable It is hard to give a simple but motivative way to introduce a normally distributed random variable. You will see later, maybe. With every brain cell we possess, let us think of a function X deFned on a sample space so that its probability density function is given by P ( X x ) = i x - 1 2 πσ exp {- ( t - μ ) 2 2 σ 2 } dt . Its density function f ( x ) = 1 2 πσ exp {- ( x - μ ) 2 2 σ 2 } has an up-side down bell shape. Jiahua Chen () Lecture 8 January-April 2011 7 / 24
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Normally distributed random variable: parameters The density function of a Normally distributed random variable is speciFed by two parameters μ R and positive σ 2 . We often use
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