Word Count : 835

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दी गई संयुक्त प्रायिकता घनत्व फलन है: [ f(x, y) = y^2 e^{-y(x+1)}; \quad x \geq 0, y \geq 0 ] --- समाश्रयण रैखिकता (Correlation) जाँचना है। ### 1. पहले, हमें ( E(X) ), ( E(Y) ), ( E(XY) ), ( Var(X) ), और ( Var(Y) ) निकालना होगा। फिर, सहसंबंध गुणांक निकाला जाएगा: [ \rho_{X,Y} = \frac{Cov(X,Y)}{\sqrt{Var(X) Var(Y)}} = \frac{E(XY) - E(X)E(Y)}{\sqrt{Var(X) Var(Y)}} ] ### 2. पहले ( f(x,y) ) की वैधता जाँचें * ( f(x,y) \geq 0 ) है ( \forall x,y \geq 0 ), सही है। * जाँचना है कि (\int_0^\infty \int_0^\infty f(x,y) dx dy = 1) है या नहीं। चलिए करें: [ \int_0^\infty \int_0^\infty y^2 e^{-y(x+1)} dx dy ] पहले अंदर की इंटीग्रल: [ \int_0^\infty e^{-y(x+1)} dx = e^{-y} \int_0^\infty e^{-yx} dx = e^{-y} \cdot \frac{1}{y} \quad \text{(यहाँ } y>0 \text{)} ] तो, [ \int_0^\infty f(x,y) dx = y^2 \cdot e^{-y} \cdot \frac{1}{y} = y e^{-y} ] अब, [ \int_0^\infty y e^{-y} dy ] यह एक ज्ञात Gamma इंटीग्रल है: [ \int_0^\infty y ____ ___ __________ _________ ____ __________ _________.
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