# Expectation and Dirac delta function

consider following expression. where $$\lambda$$ is random variable with expectation $$\mathbb{E}[\lambda] = a$$ $$= \mathbb{E}_{\lambda}\bigg[\delta \big(\tilde{x} = \lambda.x_1 + (1-\lambda)x_2 , \tilde{y} = \lambda y_1 + (1-\lambda) y_2\big) \bigg]$$

can I do this. Can I exchange dirac delta function and expectation ?

$$= \delta(\tilde{x} = \mathbb{E}[\lambda]x_1 + 1-\mathbb{E}[\lambda] x_2 , \tilde{y} = \mathbb{E}[\lambda]y_1 + 1-\mathbb{E}[\lambda] y_2)$$

$$= \delta(\tilde{x} = a x_1 + (1-a) x_2 , \tilde{y} = a . y_1 + (1-a) y_2)$$

If so, can someone provide proof of it?

• What exactly is this two-dimensional delta expression? And no, in general you can not exchange function and expectation, see Jensen's inequality on when this is partially possible. – Dr. Lutz Lehmann Sep 18 at 15:06