Question: Say I have a random variable A that is a standard normal. Say also, I have a continuous random variable B. The conditional PDF of

Say I have a random variable A that is a standard normal. Say also, I have a continuous random variable B. The conditional PDF of B given a specific value A=a, is: (1/sqrt(2*pi))*exp(-(b+(2*a))^2)/2). From this I first thought I could conclude that B is a normal as well, with mean -2*a and variance 1, because the denominator should be of the form 2*sigma^2 and if sigma^2=1, then 2*1 = 2. If that is correct, what's the procedure to find the unconditional expectation of B? I don't know whether A and B are independent or not

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