Definition:Moment (Probability Theory)/Discrete
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Definition
Let $X$ be a discrete random variable.
Then the $n$th moment of $X$ is denoted $\mu'_n$ and defined as:
- $\mu'_n = \expect {X^n}$
where $\expect {\, \cdot \,}$ denotes the expectation function.
That is:
- $\mu'_n = \ds \sum_{x \mathop \in \Omega_X} x^n \map {p_X} x$
whenever this sum converges absolutely.
It can be seen from its definition that the expectation of a discrete random variable is its first moment.
Also see the relation between the variance of a discrete random variable and its second moment.
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.3$: Moments