Please see the attached pdf file.© BrainMass Inc. brainmass.com October 10, 2019, 3:39 am ad1c9bdddf
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
Let g(x) be the density of P_x. Then we have
g(x) = ...
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.