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I have trouble with the following problem:

Gaussian random variable is defined as follows

[tex]\phi(t) = P(G \leq t)= 1/\sqrt{2\pi} \int^{t}_{-\infty} exp(-x^2/2)dx.[/tex]

Calculate the expected value

[tex]E(exp(G^2\lambda/2)).[/tex]

Hint:

Because[itex]\phi[/itex]is a cumulative distribution function,[itex]\phi(+\infty) = 1[/itex].

My attempt at solution:

I start with:

[tex] E(exp(G^2\lambda/2)) = \int^{\infty}_{-\infty}P(exp(G^2\lambda/2) \geq t)dt = \int^{\infty}_{-\infty}P(-\sqrt{2/\lambda*lnt}) \geq G \geq \sqrt{2/\lambda*lnt})dt[/tex]

[tex]=1/\sqrt{2\pi} \int^{\infty}_{-\infty}(\int^{\sqrt{2/\lambda*lnt})}_{\sqrt{2/\lambda*lnt})}e^{-x^2/2}dx)dt.[/tex]

Then my instinct would be to use Fubini theorem because I'd like to get rid of the integral of exp(-x^2/2) by [itex]\phi(+\infty) = 1[/itex].

However, because both bounds are functions of t, it wouldn't work.

Any help?

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# CDF and expectation

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