08-10-2026, 11:40 AM
Law of the unconscious statistician
Summary
The Law of the Unconscious Statistician (LOTUS) is a fundamental result in probability and statistics that allows us to calculate the expected value of a function of a random variable directly from the probability distribution of the original variable, without first finding the distribution of the transformed variable.
For a discrete random variable $X$, it states that $\operatorname{E}[g(X)] = \sum_x g(x)p_X(x)$, while for a continuous random variable it becomes $\operatorname{E}[g(X)] = \int_{-\infty}^{\infty} g(x)f_X(x),dx$. The law also extends to multiple random variables, vectors, and more general settings using measure theory, making it a powerful and widely applicable tool in probability.
ARTICLE
Summary
The Law of the Unconscious Statistician (LOTUS) is a fundamental result in probability and statistics that allows us to calculate the expected value of a function of a random variable directly from the probability distribution of the original variable, without first finding the distribution of the transformed variable.
For a discrete random variable $X$, it states that $\operatorname{E}[g(X)] = \sum_x g(x)p_X(x)$, while for a continuous random variable it becomes $\operatorname{E}[g(X)] = \int_{-\infty}^{\infty} g(x)f_X(x),dx$. The law also extends to multiple random variables, vectors, and more general settings using measure theory, making it a powerful and widely applicable tool in probability.
ARTICLE
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