Is it p(x) = 1/2 for all R? ...wait it can't be a density function right? is x is the number of coin flip and p(x) is the number of tails, how would you define it?

Is it p(x) = 1/2 for all R? ...wait it can't be a density function right? is x is the number of coin flip and p(x) is the number of tails, how would you define it?
You have two events H and T. P(H) = P(T) = 1/2. For n flips, use binomial distribution with p = 1/2.
I think you can't define a probability density function for discrete variables. Yet, you could still think of "tail" and "head" as two values of a continuous variable x and in that case the probability density function would be the sum of two Dirac functions. In that case the probability density function would be:
p(x) = 1/2 delta(x  "head") + 1/2 delta (x  "tail" )
where "head" and "tail" are the values of x you have chosen to represent "head" and "tail".
In this way, the integral (with boundaries depending on your definition of the continuous variable x) becomes 1 (1/2 + 1/2).
Correct me if I'm wrong.
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