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# lect17 - SUMS OF INDEPENDENT RANDOM VARIABLES CONDITIONAL...

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SUMS OF INDEPENDENT RANDOM VARIABLES Outline SUMS OF INDEPENDENT RANDOM VARIABLES Discrete Case Continuous Case CONDITIONAL DISTRIBUTIONS Discrete Case Continuous Case 1 / 15 Xinghua Zheng Lect 17: Sums Of Random Variables; Conditional Distributions

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SUMS OF INDEPENDENT RANDOM VARIABLES CRITERIA FOR INDEPENDENCE Random variables are independent if and only if their joint pmf/density is a product of the marginal pmf/dentities Or, equivalently, the joint pmf/density f X , Y ( x , y ) is a product of two functions with different arguments: f X , Y ( x , y ) = g ( x ) · h ( y ) , -∞ < x , y < , where 1. g ( x ) is proportional to the density/pmf of X , written as f X ( x ) g ( x ) : in fact, f X ( x ) = g ( x ) / Z -∞ g ( u ) du ( or f X ( x ) = g ( x ) / X u g ( u )); 2. h ( y ) is proportional to the density/pmf of Y , written as f Y ( y ) h ( y ) : in fact, f Y ( y ) = h ( y ) / Z -∞ h ( v ) dv ( or f Y ( y ) = h ( y ) / X v h ( v )) . 2 / 15 Xinghua Zheng Lect 17: Sums Of Random Variables; Conditional Distributions
SUMS OF INDEPENDENT RANDOM VARIABLES Discrete Case Continuous Case SUMS OF INDEPENDENT RANDOM VARIABLES Suppose X and Y are independent random variables. What’s the distribution of their sum Z := X + Y ? 1. If they’re discrete: f Z ( z ) = P [ X + Y = z ] = X x P [ X = x , Y = z - x ] = X x f X ( x ) f Y ( z - x ) . 2. If they’re continuous: cdf of Z : F Z ( z ) = P [ X + Y z ] = Z -∞ Z z - x -∞ f X , Y ( x , y ) dy dx = Z -∞ Z z - x -∞ f X ( x ) f Y ( y ) dy dx = Z -∞ f X ( x ) F Y ( z - x ) dx ; density of Z : f Z ( z ) = F 0 Z ( z ) = Z -∞ f X ( x ) f Y ( z - x ) dx . 3 / 15 Xinghua Zheng Lect 17: Sums Of Random Variables; Conditional Distributions

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SUMS OF INDEPENDENT RANDOM VARIABLES Discrete Case Continuous Case EXAMPLE: SUM OF INDEPENDENT BINOMIALS Suppose X B ( n , p ) , Y B ( m , p ) , and they are independent, then their sum Z = X + Y . Verify this fact using the previous formula: for any integer 0 k m + n , f Z ( k ) = min ( n , k ) X i = 0 ± n i ² p i q n - i · ± m k - i ² p k - i q m - k + i = = . 4 / 15 Xinghua Zheng Lect 17: Sums Of Random Variables; Conditional Distributions
SUMS OF INDEPENDENT RANDOM VARIABLES Discrete Case Continuous Case EXAMPLE: SUM OF INDEPENDENT POISSONS Recall: if X Poisson ( λ 1 ) , Y Poisson ( λ 2 ) , and they are independent, then their sum Z = X + Y .

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lect17 - SUMS OF INDEPENDENT RANDOM VARIABLES CONDITIONAL...

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