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?

  • $\begingroup$ 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. $\endgroup$ – Dr. Lutz Lehmann Sep 18 at 15:06

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