Btukfyl

1 1 Suppose we have a random sample $X_1, X_2, ldots, X_n$ from exponential $~(β >0)$ $text{i.e. }f(xmid β) = {1/β} ~e^{−x/β}$ and a random sample $~Y_1, Y_2, ldots, Y_n$ from exponential $~(⍺ >0)$ and assume both sample are independent. let $~~ θ = P(X_1 < Y_1)$ Find the MVUE of $~θ~~$ for $n=2$ . So, first I calculate the $θ=int_0^ ∞int_x^infty frac{1}{β} ~e^{−x/β}~~ frac{1}{⍺ } ~e^{−y/⍺} , dy , dx= fracalpha {alpha+beta}$ Then since $f(X,Y) =f(X)cdot f(Y) =f(X_1)cdot f(X_2)cdot f(Y_1)cdot f(Y_2)$ belongs to exponential family then $(x_1+x_2, y_1+y_2)$ is complete sufficient statistics. $x_1+x_2sim operatorname{Gamma}(2,beta)$ and $y_1+y_2sim operatorname{Gamma}(2,alpha).$ Now, I am stuck, any help please. mathematical-statistics