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## Homework Statement

Consider the bivariate vector random variable ##(X,Y)^T## which has the probability density function $$f_{X,Y}(x,y) = \theta xe^{-x(y+\theta)}, \quad x\geq 0, y\geq 0 \; \; \text{and} \; \; \theta > 0.$$

I have shown that the marginal distribution of ##X## is ##f_X(x|\theta) = \theta e^{-\theta x}, \quad x\geq 0 \; \; \text{and} \; \; \theta > 0.##

My question is why do these two distributions have the same maximum likelihood estimator ##\hat{\theta} = \frac{1}{\bar{x}}??##