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Math Help - Expectation

  1. #1
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    Expectation

    Hey,

    I have given a normally distributed random variable $X\stackrel{d}{=}\mathcal{N}(\mu,\sigma^2)$ and want to compute the following expectation

    $\mathbb{E}[X \mathbb{1}_{\{X\geq 1\}}]$

    where $\mathbb{1}$ denotes the indicator function.

    I think that it is not possible to compute the corresponding integral explicitly. Am I right?

    Thanks in advance!
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  2. #2
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    Re: Expectation

    Hey Juju.

    Can you find the random variable corresponding to X >= 1 where X has that Normal distribution?

    Hint: Use the conditional distribution of X|X>=1 where P(A|B) = P(A and B)/P(B).
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  3. #3
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    Re: Expectation

    Thank you for your reply,
    unfortunately I do not understand what you mean by "corresponding random variable". Of course I can determine the distribution function of X \mathds{1}_{\{X\geq 1\}} and the expectation is given by
    \int\limits_1^\infty (2 \pi \sigma^2)^{-\frac{1}{2}}xe^{-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}}dx

    But I cannot give this integral in a closed form.
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  4. #4
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    Re: Expectation

    You should be able to evaluate the integral: (Hint: use a substitution v = (x-mu)^2 and change from dx to dv).

    Also when I say corresponding random variable I just mean the random variable that is represented algebraically with respect to the limits of the integrand.

    For example I can have Y = X + 3 and from this specify a conditional probability of P(Y|X) if I know the density function for the random variable X.

    Now similarly, you can have limits in the integration that correspond with a particular algebraic expression but they may not have an easy interpretation algebraically (i.e. might be a lot more complex than the relation Y = X + 3).
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