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    CDF, Expected Value and Variance

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    Solution Preview

    Note: There appears to be an error in the statement of the problem. The factor 2/sqrt(pi) needs to be replaced by sqrt(2/pi) in order for P_1 and P_x to be probability distributions.

    1. We are given

    P_x = 1/2(delta_0 + P_1),

    where P_1 has density given by

    f(x) = sqrt(2/pi) e^(-x^2/2) H(x)

    where H(x) is the Heaviside step function, i.e.

    H(x) = 1 if x > 0
    0 otherwise

    Let g(x) be the density of P_x. Then we have

    g(x) = ...

    Solution Summary

    We compute the cumulative distribution function (CDF) for a given probability distribution f(x), as well as the expected value and variance of x for this distribution.