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Jun 27, 15 · 3 FY(y) = P(Y ≤ y) is the probability that the random variable Y is less than or equal to a given real value y This function is then known as the cumulative distribution function For a continuous random variable it is the integral of the probability density function up to y, while for a discrete random variable it is the partial sum up toP n i=1 a iY i = Ee P n i=1 ta iY i = Yn i=1 Eeta iY i = Yn i=1 m Y i (ta i) = Yn i=1 eµta i 1 2 σ 2(ta i) = e P n i=1 µta i P n i=1 1 2 σ 2(ta i) = e(P n i=1 µa i)t1 2 P n i=1 a 2σ t which is the mgf of N(P n i=1 a iµ, P n i=1 a 2 i σ 2) So, U has a normal distribution with mean P n i=1 a iµ and variance P n i=1 2 i σ 2 657 Y¥ å É Û å « í £ î « È µ ¡ &¾ _%&1/ K Z C d M D P Â M b _ \0 Z K r W S > ½ /$× ^/ · í* < > ¥ å É Û å « $ ( c ¥ å É Û å «4* o @ 8 Â K S Ý G P*( /

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