Raw Moment of Bernoulli Distribution
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Theorem
Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$.
Let $n$ be a strictly positive integer.
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:
- $\expect {X^n} = p$
Proof 1
From the definition of expectation:
- $\ds \expect {X^n} = \sum_{x \mathop \in \Img X} x^n \map \Pr {X = x}$
From the definition of the Bernoulli distribution:
- $\ds \expect {X^n} = 1^n \times p + 0^n \times \paren {1 - p} = p$
$\blacksquare$
Proof 2
By Moment Generating Function of Bernoulli Distribution, the moment generating function $M_X$ is given by:
- $\map {M_X} t = q + p e^t$
By Moment in terms of Moment Generating Function:
- $\expect {X^n} = \map {M^{\paren n}_X} 0$
By Derivative of Exponential Function:
- $\map {M^{\paren n}_X} t = p e^t$
Setting $t = 0$:
\(\ds \expect {X^n}\) | \(=\) | \(\ds p e^0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p\) | Exponential of Zero |
$\blacksquare$