Expectation of Bernoulli Distribution/Proof 4
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Theorem
Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$.
Then the expectation of $X$ is given by:
- $\expect X = p$
Proof
From Moment Generating Function of Bernoulli Distribution, the moment generating function of $X$, $M_X$, is given by:
- $\map {M_X} t = q + p e^t$
where $q = 1 - p$.
By Moment in terms of Moment Generating Function:
- $\expect X = \map {M_X'} 0$
We have:
\(\ds \map {M_X'} t\) | \(=\) | \(\ds \frac \d {\d t} \paren {q + p e^t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p e^t\) | Derivative of Constant, Derivative of Exponential Function |
Setting $t = 0$ gives:
\(\ds \expect X\) | \(=\) | \(\ds p e^0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p\) | Exponential of Zero |
$\blacksquare$