Problem 2: expectation of minimum

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Say we have X ~ Uniform(0, 1) and Y ~ Uniform(0, 1). What is the expected value of the minimum of X and Y?


It is not stated that $X$ and $Y$ are independent but we will assume it. Let $Z = \min \{ X,Y\}$, $F_X,F_Z$ be the cumulative distribution of $X,Z$ respectively and $f_X,f_Z$ the density of $X,Z$ respectively. Note that

\[F_Z(t) = \mathbb{P}(Z \leq t) = 1 - \mathbb{P}(Z \geq t),\]


\[\mathbb{P}(Z \geq t) = \mathbb{P}(X \geq t \text{ and } Y \geq t) = (1 - F_X(t))^2.\]

Differentiating, we obtain the density of $Z$:

\[f_Z(t) = 2(1-F_X(t))f_X(t).\]

Recalling that the density of a uniform random variable is constant in its support and the cumulative distribution is linear in $t$. We obtain then

\[f_Z(t) = 2(1-t)\]

for $t\in (0,1)$. Then the expected value of $Z$ is

\[\mathbb{E}(Z) = \int_{0}^1 f_Z(t) t dt = \dfrac{1}{3}.\]

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