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Unformatted text preview: MA2216/ST2131 Probability Notes 5 Distribution of a Function of a Random Variable We have seen that an “affine transformation” of a normal r.v. is still normally distributed. To be precise, let X ∼ N ( μ,σ 2 ). Put Y = aX + b . Then, it has been pointed out in § 8.8 of Notes 4 that, for a 6 = 0 , Y ∼ N ( aμ + b, a 2 σ 2 ) . Derivation. Recall that f X ( x ) = 1 √ 2 πσ 2 e ( x μ ) 2 2 σ 2 . To determine the distribution of Y , find F Y , the d.f. of Y , first. Consider the case a > 0 first. F Y ( y ) = IP( Y ≤ y ) = IP( aX + b ≤ y ) = IP( X ≤ y b a ) = F X y b a ¶ . To identify the distribution of Y , we need to find its p.d.f., f Y , which is the derivative of F Y : for y ∈ IR, f Y ( y ) = d dy F Y ( y ) = d dy F X y b a ¶ = f X y b a ¶ · 1 a = 1 a · 1 √ 2 πσ 2 e ( y b a μ ) 2 2 σ 2 = 1 p 2 π ( aσ ) 2 e ( y aμ b ) 2 2( aσ ) 2 . We recognize that it is the p.d.f. of N ( aμ + b, a 2 σ 2 ). 1 If a < 0, then F Y ( y ) = IP X ≥ y b a ¶ = 1 F X y b a ¶ . Hence, f Y ( y ) = 1 a · 1 √ 2 πσ 2 e ( y b a μ ) 2 2 σ 2 = 1 p 2 π ( aσ ) 2 e ( y aμ b ) 2 2( aσ ) 2 , which is obviously the p.d.f. of the normal distribution N ( aμ + b, a 2 σ 2 ), and therefore, we are done. In particular, if we set Z = X μ σ , then Z is a standard normal r.v., i.e., Z ∼ N (0 , 1). This is perhaps the most frequently used transformation. In the following, we will see a few additional examples on the derivation of the distribution of a transformed r.v. 2 1. Example. Let Z ∼ N (0 , 1). What are the d.f. and p.d.f. of Y , where Y = Z 2 ? Note that, Y is known in the literature as chisquare ( χ 2 ) r.v. of degree 1. Sol. First note that Y takes nonnegative values. Obviously, Y is con tinuous, and hence we determine its c.d.f. first. For y > 0, we have F Y ( y ) = IP( Y ≤ y ) = IP( Z 2 ≤ y ) = IP( √ y ≤ Z ≤ √ y ) = 1 √ 2 π Z √ y √ y e z 2 / 2 dz, = 2 √ 2 π Z √ y e z 2 / 2 dz, hence f Y ( y ) = dF Y ( y ) dy = 2 √ 2 π 1 2 √ y e y/ 2 = y 1 / 2 e y/ 2 √ 2 π = ( 1 2 ) 1 2 Γ( 1 2 ) y 1 2 1 e 1 2 y . We conclude that F Y ( y ) = , for y ≤ 0, 2 √ 2 π R √ y e u 2 / 2 du, for y > 0. and f Y ( y ) = , for y ≤ 0, ( 1 2 ) 1 2 Γ( 1 2 ) y 1 2 1 e 1 2 y , for y > 0. Remark. Do you recognize the distribution of the above Y ? In fact, the distribution of Z 2 with Z ∼ N (0 , 1) is Γ( 1 2 , 1 2 ). Thus, χ 2 1 = Γ 1 2 , 1 2 ¶ . This is a very useful result, which will be quoted often throughout. 3 Addendum. * Refer to § 1.3 of Notes 7, in which the following property will be established: If X i , i = 1 , 2 ,...,n are independent gamma r.v.’s with respective pa rameters ( α i ,λ ) , i = 1 , 2 ,...,n , then n X i =1 X i ∼ Γ ˆ n X i =1 α i , λ !...
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This note was uploaded on 03/19/2012 for the course SCIENCE ST2131 taught by Professor Forgot during the Fall '08 term at National University of Singapore.
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