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Sums of Random Variables Probability Density of a Sum of Random Variables Let Z = X + Y . Then for Z to be less than z , X must be less than z − Y . Therefore, the CDF for Z is FZ ( z ) = ∞ z− y ∫ ∫ f ( x, y ) dxdy XY −∞ −∞ If X and Y are independent, ⎛ z− y ⎞ FZ ( z ) = ∫ fY ( y ) ⎜ ∫ f X ( x ) dx ⎟dy −∞ ⎝ −∞ ⎠ and it can be shown that ∞ fZ ( z ) = ∞ ∫ f ( y ) f ( z − y ) dy = f ( z ) ∗ f ( z ) ← Convolution Y −∞ X Y X Moment Generating Functions The moment-generating function Φ X ( s ) of a CV random variable X is defined by Φ X ( s ) = E ( e sX ) = ∞ ∫ f X ( x ) e sx dx. −∞ Φ X ( s ) = L ⎡⎣f X ( x ) ⎤⎦ s →− s where L ⎡⎣f X ( x ) ⎤⎦ is the Laplace transform of f X ( x ) ∞ d Φ X ( s ) ) = ∫ f X ( x ) xe sx dx ( ds −∞ ∞ ⎡d ⎤ = ∫ x f X ( x ) dx = E ( X ) ⎢⎣ ds ( Φ X ( s ) ) ⎥⎦ s →0 −∞ n ⎡ ⎤ d n E ( X ) = ⎢ n ( Φ X ( s ) )⎥ ⎣ ds ⎦ s →0 Moment Generating Functions Example Let X and Y be exponential random variables with PDF's ⎧λ X e − λ X x , x > 0 ⎧λY e − λY y , y > 0 fX ( x) = ⎨ and fY ( y ) = ⎨ , otherwise , otherwise ⎩0 ⎩0 Find the PDF of Z = X + Y . f Z ( z ) = f X ( z ) ∗ fY ( z ) The moment-generating function of Z is Φ Z ( s ) = Φ X ( s ) ΦY ( s ) . ΦX (s) = λX λX − s , ΦY ( s ) = λY λY − s ⇒ ΦZ ( s) = λX λY λ X − s λY − s Moment Generating Functions Example ⎛ λX λY ⎞ f Z ( z ) = L ( Φ X ( − s ) ΦY ( − s ) ) = L ⎜ ⎟ + λ + λ s s ⎝ X Y ⎠ −1 λ X λY ⎛ λ X λY ⎜ λ X − λY −1 λY − λ X fZ ( z ) = L ⎜ + s + λY ⎜ s + λX ⎜ ⎝ −1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ λ X λY − λY z ⎧ λ X λY − λX z , z>0 e e + ⎪ λX − λY f Z ( z ) = ⎨ λY − λX ⎪0 , otherwise ⎩ Moment Generating Functions The moment-generating function Φ X ( s ) of a DV random variable X is defined by Φ X ( s ) = E ( e sX ) = ⎡d ⎤ = E(X ) ⎢⎣ ds Φ X ( s ) ⎥⎦ s →0 ∞ ∑ n =−∞ P ( X = n ) e sn = ∑ x∈S X n ⎡ ⎤ d n E ( X ) = ⎢ n ( Φ X ( s ) )⎥ ⎣ ds ⎦ s →0 PX ( x ) e sx . Moment Generating Functions Example Let X be a Poisson random variable. Its moment-generating function is ( ) α e s −1 ΦX (s) = e . Find its mean and variance. s ⎡d ⎤ s α ( e −1) ⎤ ⎡ = αe e =α E ( X ) = ⎢ Φ X ( s )⎥ ⎥⎦ s →0 ⎣ ds ⎦ s →0 ⎢⎣ 2 s s ⎡ ⎤ d s α ( e −1) ⎤ 2 2 2 s α ( e −1) ⎡ = α e e E ( X ) = ⎢ 2 Φ X ( s )⎥ +αe e = α 2 +α ⎥⎦ s →0 ⎣ ds ⎦ s →0 ⎢⎣ Var ( X ) = α 2 + α − α 2 = α The Central Limit Theorem If N independent random variables are added to form a resultant N random variable Z = ∑ X n then n =1 f Z ( z ) = f X1 ( z ) ∗ f X 2 ( z ) ∗ f X 2 ( z ) ∗" ∗ f X N ( z ) and it can be shown that, under very general conditions, the pdf of a sum of a large number of independent random variables with continuous pdf’s approaches a Gaussian pdf regardless of the shapes of the individual pdf’s. The Central Limit Theorem Multivariate Probability Density FX1 , X 2 ,", X N ( x1 , x2 ," , xN ) ≡ P ( X 1 ≤ x1 ∩ X 2 ≤ x2 ∩ " ∩ X N ≤ xN ) 0 ≤ FX1 , X 2 ,", X N ( x1 , x2 ," , xN ) ≤ 1 , − ∞ < x1 < ∞ , " , − ∞ < xN < ∞ FX1 , X 2 ,", X N ( −∞," , −∞ ) = FX1 , X 2 ,", X N ( −∞," , xk ," , −∞ ) = FX1 , X 2 ,", X N ( x1 ," , −∞," , xN ) = 0 FX1 , X 2 ,", X N ( +∞," , +∞ ) = 1 FX1 , X 2 ,", X N ( x1 , x2 ," , xN ) does not decrease if any number of x ' s increase FX1 , X 2 ,", X N ( +∞," , xk ," , +∞ ) = FX k ( xk ) Multivariate Probability Density f X1 , X 2 ,", X N ( x1 , x2 ," , xN ) = ∂N ∂ x1∂ x2 " ∂xN FX1 , X 2 ,", X N ( x1 , x2 ," , xN ) f X1 , X 2 ,", X N ( x1 , x2 ," , xN ) ≥ 0 , − ∞ < x1 < ∞ , " , − ∞ < xN < ∞ ∞ ∞ ∞ ∫"∫ ∫ f −∞ −∞ −∞ X1 , X 2 ,", X N FX1 , X 2 ,", X N ( x1 , x2 ," , xN ) = f X k ( xk ) = ∞ −∞ −∞ x2 x1 ∫"∫ ∫ f −∞ ∞ ∞ ∫"∫ ∫ f −∞ xN ( x1 , x2 ," , xN ) dx1dx2 " dxN X1 , X 2 ,", X N −∞ −∞ X1 , X 2 ,", X N =1 ( λ1 , λ2 ," , λN ) d λ1d λ2 " d λN ( x1 , x2 ," , xk −1 , xk +1 ," , xN ) dx1dx2 " dxk −1dxk +1 " dxN P {( X 1 , X 2 ," , X N ) ∈ R} = ∫ " ∫∫ f X1 , X 2 ,", X N ( x1 , x2 ," , xN ) dx1dx2 " dxN E ( g ( X 1 , X 2 ," , X N ) ) = R ∞ ∞ ∫ ∫ g ( x , x ," , x ) f 1 −∞ −∞ 2 N X1 , X 2 ,", X N ( x1 , x2 ," , xN ) dx1dx2 " dxN