# What's the probability density function of a fair coin flip?

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• July 21st, 2013, 11:03 PM
EdmureTully
What's the probability density function of a fair coin flip?
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?
• July 22nd, 2013, 05:34 PM
mathman
You have two events H and T. P(H) = P(T) = 1/2. For n flips, use binomial distribution with p = 1/2.
• July 23rd, 2013, 02:51 AM
fred91
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.
• July 23rd, 2013, 09:20 PM
mathman
Quote:

Originally Posted by fred91
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.

You are correct in asserting you can't have a density function for discrete random variables. However you are making things unnecessarily complicated. In these cases simply work with the distribution function.