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239概率困难derivationlong

Ratio of Independent Standard Normals is Cauchy

题目

Let Z1,Z2Z_1, Z_2 be independent N(0,1)N(0,1) random variables. Define R=Z1/Z2R = Z_1 / Z_2.

(a) Write the joint PDF of (Z1,Z2)(Z_1, Z_2) and use the transformation (Z1,Z2)↦(R,Z2)=(Z1/Z2, Z2)(Z_1, Z_2) \mapsto (R, Z_2) = (Z_1/Z_2,\, Z_2) to derive the joint PDF of (R,Z2)(R, Z_2).

(b) Integrate out Z2Z_2 to obtain the marginal PDF of RR.

(c) Identify the distribution of RR.

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