TITLE: Can a Dirac delta function be a probability density function of a random variable?
QUESTION [17 upvotes]: Can the Dirac delta function (or distribution) be a probability density function of a random variable. To my knowledge, it seem to satisfy the conditions. To my interpretation getting a positive real number as the outcome is 1 and that for a negative real number is zero. I wonder what could be the expected value. My question is, whether it is a valid probability density function of a random variable.

REPLY [2 votes]: I was thinking about linear transformation of a zero mean gaussian random variable X. Defining random variable Y=aX, one can easily shows that the probability density function of Y is: $$ p_Y(y)=\frac{1}{|a|}.p_X(y/a)$$ It can be showed that as "a" tends to zero, the $$ p_Y(y)\to\delta (y)$$ Therefore, in my opinion, Dirac delta function can be used to describe the PDF of such a random variable.