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lecture12 - Stat 302, Introduction to Probability Jiahua...

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Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 12 January-April 2011 1 / 21
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Sum of two continuous random variables Suppose X and Y have joint pdf given by f ( x , y ) . Then the cdf of X + Y is given by P ( X + Y a ) = i y = b i x = a y x = f ( x , y ) dx B dy . Taking derivative with respect to a , one gets the pdf of X + Y f X + Y ( a ) = i y = f ( a y , y ) dy . When X and Y are independent, f X + Y ( a ) = i y = f X ( a y ) f Y ( y ) dy . Jiahua Chen () Lecture 12 January-April 2011 2 / 21
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Sum of two continuous random variables Suppose X and Y have joint pdf given by f ( x , y ) = λ 2 exp ( λ x λ y ) I ( x > 0, y > 0 ) Then the pdf of X + Y is given by (for a > 0) f X + Y ( a ) = i y = f X ( a y ) f Y ( y ) dy = λ 2 i a y = 0 exp ( λ a ) dy = λ 2 a exp ( λ a ) . Let us write T = X + Y and its pdf as f T ( t ) = λ 2 t exp ( λ t ) I ( t > 0 ) . Jiahua Chen () Lecture 12 January-April 2011 3 / 21
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Sum of two continuous random variables Suppose X 1 , X 2 , . . . , X m are independent and have identical exponential distribution with intensity parameter λ . The pdf of T = X 1 + X 2 + · · · + X m is given by f T ( t ; m , λ ) = λ m Γ ( m ) t m 1 exp ( λ t ) for t > 0. We say that T has Gamma distribution with m degrees of freedom and scale parameter λ . Jiahua Chen () Lecture 12 January-April 2011 4 / 21
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Sum of Gamma random variables Apparently, when X and Y are independent and have Gamma( m 1 , λ ) and Gamma( m 2 , λ ) distributions, then X + Y has Gamma( m 1 + m 2 , λ ) distribution. Jiahua Chen () Lecture 12 January-April 2011 5 / 21
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Sum of Gamma random variables Apparently, when X and Y are independent and have Gamma( m 1 , λ ) and Gamma( m 2 , λ ) distributions, then X + Y has Gamma( m 1 + m 2 , λ ) distribution. Take note that the above conclusion is obtained when X and Y share the same scale parameter λ . Jiahua Chen () Lecture 12 January-April 2011 5 / 21
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Motivation for exponential distribution When a product does not wear and tear, then its survival time is best modeled by an Exponential distribution, as this distribution has memoryless property . If the scale 7+ earthquakes occur as a (homogeneous) Poisson
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lecture12 - Stat 302, Introduction to Probability Jiahua...

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