A probability distribution gives the probability of a random variable being (exactly|in_the_range_of) some value.

Expected Value

The expected value ⟨X⟩ of a random variable X is the mean, the average, the value of the argument you have to supply to the probability distribution to get the maximum of the function.

⟨X⟩:=\sum\limits_i x_i⋅f(x_i)

where f is the probability mass function.

Or for continuous distributions:

⟨X⟩:=∫\limits_{-∞}^∞ x⋅ϕ(x)⋅dx

where ϕ is the probability density function.


The variance V, given some data points x⃗, is:


Sometimes, the variance of the entire set (not just the sample) is:

V_s:=÷{1}{n-1}⋅\sum\limits_{i=1}^n (⟨X⟩-x_i)²

where n is the maximum index of x⃗.