Joint and Marginal Densities.
- Thread starter Actuarial_Deepika
- Start date
- Tags cdf data analysis joint density marginal density probability
I think the joint CDF should be
\( F(x, y) = (1 - e^{-\alpha x})(1 - e^{-\beta y}), x, y > 0, \alpha, \beta > 0 \)
If you are familiar with exponential distribution, you can directly see that this
is the joint CDF of two independent exponential distribution.
Of course you may check with the following way easily:
The joint pdf \( f(x, y) = \frac {\partial^2 F(x, y)} {\partial x \partial y}
= \frac {\partial^2 F(x, y)} {\partial y \partial x} \)
The marginal pdf
\( f(x) = \int_0^{+\infty} f(x, y)dy, f(y) = \int_0^{+\infty}f(x, y)dx \)
\( F(x, y) = (1 - e^{-\alpha x})(1 - e^{-\beta y}), x, y > 0, \alpha, \beta > 0 \)
If you are familiar with exponential distribution, you can directly see that this
is the joint CDF of two independent exponential distribution.
Of course you may check with the following way easily:
The joint pdf \( f(x, y) = \frac {\partial^2 F(x, y)} {\partial x \partial y}
= \frac {\partial^2 F(x, y)} {\partial y \partial x} \)
The marginal pdf
\( f(x) = \int_0^{+\infty} f(x, y)dy, f(y) = \int_0^{+\infty}f(x, y)dx \)