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Unformatted text preview: Continuous distributions 1. X & GAM( &;Â¡ ): Gamma distribution with parameters & > and Â¡ > density f X ( x ) = ( x & ) Â¡ e & x & x &( & ) , x > mean E [ X ] = &Â¡ variance V ar ( X ) = &Â¡ 2 m.g.f. M X ( s ) = & 1 1 & Â¡s Â¡ & , s < 1 Â¡ Â¡ For the gamma distribution de&ned above, & the parameter Â¡ is a scale parameter & the parameter & is a shape parameter Â¡ In general, there is no closed-form expression for the cumulative distribution function of a gamma distribution. However, for a sub-class of the gamma distributions, namely those for which the shape parameter & is an integer, a closed-form expression can be found via integration by parts. This sub-class of gamma distributions is known as the class of Erlang distributions . For X & GAM( n;Â¡ ) with n a positive integer, F X ( x ) = Z x & y Â¡ Â¡ & e & y & y & ( & ) dy = 1 Â¢ n & 1 X k =0 & x Â¡ Â¡ k e & x & k ! , x > . 2. X & EXP( Â¡ ): Exponential distribution with parameter Â¡ > density f X ( x ) = 1 Â¡ e & x & , x > c.d.f. F...
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This note was uploaded on 11/23/2010 for the course ACTSC act431 taught by Professor Na during the Spring '09 term at Waterloo.
- Spring '09