λ²e^(-λn) × 1 / y² is the joint probability density function of x1 and x2.
If x1 and x2 are two independent random variables,
The joint probability density function is equal to the product of the marginal distribution functions of two random variables if they are independent.
The joint probability density function of x1 and x2 is,
[tex]f(x1,x2) =[/tex] λ² e^(-λ(x1 + x2))
consider
[tex]y1 = x1 + x2 ; \\y2 = e^{x1}[/tex]
The joint probability density function of y1 and y2 is,
[tex]f(y1,y2) = f(x1,x2) . |J(x1,x2) |^{-1}[/tex]
The Jacobean transformation on solving determinant is J(x1,x2)
[tex]J(x1,x2) =[/tex] 1(0) - 1(eˣ¹)
= - (eˣ¹)
Now substituting values of y1 and y2
f(y1, y2) = λ² e^(-λ(x1 + x2)) . ([tex]{\frac{1}{e^{x1} } }[/tex])
since
[tex]y1 = x1 + x2 ; \\y2 = e^{x1}[/tex]
= λ²e^(-λn) × 1 / y²
Hence λ²e^(-λn) × 1 / y² is the joint probability density function of y1 and y2.
To learn more about joint probability density function from the given link
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