Word Count : 835

---

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 _____ ______ ___ ____ _________ ______ ______ ______.
_____ __________ ___ _________ _______ _____ ____ _________ ___ _______ ______.
____ ____ _______ __________ _________.
___ ________ __________ ____ ____.
____ ______ _________ __________ ___ _________ ______.
_________ __________ _____ _______ ________.
______ ________ _____ _______ _______ _____ ___.
______ ___ ____ _________ ________ ________ ____ ____.
__________ ____ _______ ________ __________ _____ ______ _______ ________ _____ _________ _______.
_______ ________ _______ _______ ___ _______ _____.
________ ________ _______ __________ ______.
____ __________ _______ _____ ____ ________ _____ ____.
____ ______ ______ _________ _____ _______ ___ _________.
______ _______ ______ _______ __________ ___ __________ _______ ________ ______.
_________ ___ _______ ______ _____ ______.
_______ ______ ____ __________ _____.
______ __________ __________ _______ _____ ____ ________ _________.
_____ _________ ________ _______ __________ __________ _______ _________ ____ _______ __________ ______.
_________ __________ ________ _____ ___ _________ ______ _________ ______.
_______ _________ _____ ____ _______ _________ _____ __________ __________ _____.
_______ _______ _______ ___ ____ _______ ________ ___ __________ ____ __________.
_________ ______ __________ _________ ___.
_______ _________ ______ _______ ___ ______ ______ ___ ___ ____ ___ _________.
______ __________ _________ ________ _____ ____ __________ _________ _____ ____.
________ _______ ______ _______ _____ ________ _________ ____ _____ ___.
_________ ______ __________ ______ ___ ___ _______ _______ _______ ________ ______.
_____ ________ _______ ______ ___ _____ ______ __________ _______ _______ _____.
_______ ______ ___ ________ _________.
_____ ______ _________ _____ _________ ____ _______ ________ ________.
______ _______ ______ ________ _______ ________ ______ _____ _______.
_________ __________ _______ ____ ________ _________ ____ ____.
______ _____ ___ _______ ______ ________.
________ ____ ________ _______ __________ ___ ____ ______ __________ _____.
________ ______ __________ _______ __________ ______ ______ ___ ______ _____ _______ ____.
___ ___ ______ ___ _____ __________.
___ ______ __________ ________ ______ ___ ____ ____ _________ _______ _____ __________.
_____ ____ ______ ____ _______ _______ _______ _______ _______ _______ ______ ______.
______ ______ _________ __________ ____ ______ ____ ___ ___ _______ ________ _________.
____ _________ _______ _____ ___ ____ _____ __________ _____ _______ _________ _________.
________ ____ ________ _______ ____ _______ _____ _______ ____ ____ _____.
_______ __________ ____ ____ __________ _____ ______ ________ ____ ______.
_____ _____ ___ _________ _____ ______ ________ _____ __________ ________.
____ __________ ___ _______ ______ __________ ____.
______ _________ _______ _________ _________ ________ _____ ________ ________ ________.
____ _________ _______ __________ ______ _______ __________ ______ _______ _____ ___ ______.
________ ___ _____ ______ ___ _____ _______ _________ __________ _________ _________ ___.
__________ __________ _____ _________ __________ _______ ________.
______ _____ ___ ________ ______ ___ ________ _________ ________ ________ ___ __________.
__________ __________ ____ _________ ___ _________ ________ ____.
________ ___ _________ _________ ____ __________ ________ ____ _______ ___ _________.
_________ _________ _________ __________ ____ ____ _________ __________ __________ _______ _____.
__________ ________ ___ ___ _______.
_____ _______ _________ ______ __________ ____ _______ _______.
_________ _______ ____ ____ ________ _________ ___ ____ ________ ____.
_____ ___ ________ ______ _______ _________ _______ _____ _________ ___.
_________ _______ ______ ____ ____ _______ ______ ________.
___ _________ ___ _____ ____ _______ ___ __________ ____.
_____ _________ ________ _________ ________ ____ ________ ___ _____.
_______ _________ _________ _____ _________.
___ ________ ________ ________ _______ ______ _______ ______ ________ ________ __________.
_________ ________ ____ _______ ____ ____ __________ _______ _____.
___ ___ ___ ______ ________ _______ ___ _______ _________ ______.
__________ ________ ___ _________ ____ ___.
_____ _________ __________ ______ __________ _____ ___ ___.
________ ________ ___ _____ _________ __________.
__________ ___ ____ _______ ____ _________ ___.
______ ___ ___ ___ __________ _________ ______ ___ _________.
____ __________ ___ _____ _____ ____ _________ ______ __________.
______ ________ ____ _____ ______.
_______ _____ ________ ____ __________ __________ _____ ________ ______.
_________ _________ ___ _________ ____ ________ ________ ___.
_________ ___ ________ ___ ______.
_____ _________ ________ _______ _____ _________ ____ ______ ___ _______ _________ ____.
________ ______ ___ _______ _____ _____.
__________ _________ ______ _____ ______ ____ _____ _________ ____ ___ __________.
_______ ________ __________ ___ _________ ___ _________ __________ ___ _______ ______ ________.
_______.
Get Full Answer on WhatsApp