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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). The unconditional PDF of a A is just that of the standard normal distribution. The unconditional PDF of B can be calculated from the previous values. I know that I can get the joint PDF if I have at least one conditional PDF - which I have - and the individual PDFs. Is it correct, that to get the joint PDF I simply multiply the PDF of A with the conditional PDF of B given A? Having the joint PDF allows me to calculate cov(A,B), is that correct

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