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Unformatted text preview: MA2216/ST2131 Probability Notes 7 Sums of Independent Random Variables Very often we are interested in the sum of independent random variables. When X and Y are independent, we would like to know the distribution of X + Y . In the following, we will deal with continuous as well as discrete distributions, and derive some very important properties. § 1. Continuous, Independent R.V.’s Under the assumption of independence of X and Y , we have f X,Y ( x,y ) = f X ( x ) f Y ( y ) , for x,y ∈ IR . In order for us to derive the p.d.f. of X + Y , we need to find the distri bution function of X + Y first. Then, for w ∈ IR, F X + Y ( w ) = IP( X + Y ≤ w ) = Z Z { ( x,y ): x + y ≤ w } f X,Y ( x,y ) dxdy = Z Z { ( x,y ): x + y ≤ w } f X ( x ) f Y ( y ) dxdy = Z ∞∞ •Z w y∞ f X ( x ) dx ‚ f Y ( y ) dy = Z ∞∞ F X ( w y ) f Y ( y ) dy. Similarly, one can show that F X + Y ( w ) = Z ∞∞ •Z w x∞ f Y ( y ) dy ‚ f X ( x ) dx = Z ∞∞ F Y ( w x ) f X ( x ) dx. Summary: F X + Y ( w ) = Z ∞∞ F X ( w y ) f Y ( y ) dy = Z ∞∞ F Y ( w x ) f X ( x ) dx. (1 . 1) 1 It then follows that the p.d.f., f X + Y ( w ), of W = X + Y is given by f X + Y ( w ) = d dw F X + Y ( w ) = Z ∞∞ d dw F X ( w y ) f Y ( y ) dy = Z ∞∞ f X ( w y ) f Y ( y ) dy. Summary: f X + Y ( w ) = Z ∞∞ f X ( w y ) f Y ( y ) dy = Z ∞∞ f X ( x ) f Y ( w x ) dx. (1 . 2) Remark. Let g and h be two “ nice ” functions on IR (such as integrable functions). The convolution of g and h , denoted by g * h , is defined to be g * h ( z ) def. = Z ∞∞ g ( z y ) h ( y ) dy. It can be shown that g * h is also given by g * h ( z ) = Z ∞∞ g ( x ) h ( z x ) dx. In other words, the convolution * is a commutative operation , i.e., g * h = h * g. Thus, the p.d.f. f X + Y is the convolution of the p.d.f.’s f X and f Y . We now turn to a few examples. 2 1. Sum of 2 Independent Exponential Random Variables. Suppose that X and Y are independent with a common exponential distribution with parameter λ > 0. Find the p.d.f. of X + Y . Derivation. Note that X + Y takes values in (0 , ∞ ). For w ≤ 0, it then follows that f X + Y ( w ) = 0 . For 0 < w < ∞ , f X + Y ( w ) = Z ∞∞ f X ( w y ) f Y ( y ) dy = Z w λe λ ( w y ) · λe λ y dy = Z w λ 2 e λ w dy = λ 2 w e λ w . In summary, f X + Y ( w ) = λ 2 w e λ w , for 0 < w < ∞ , , elsewhere. Obviously, the distribution of X + Y is a gamma distribution of param eters 2 and λ . Such a result can be generalized to a sum of independent gammas having the same second parameter, which is to be dealt with next. 3 2. Sum of 2 Independent Gammas. Assume that X ∼ Γ( α,λ ) and Y ∼ Γ( β,λ ), and X and Y are mutually independent. Then, X + Y ∼ Γ( α + β, λ ) ....
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 Fall '08
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