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Let $X_i$ be a uniformly distributed random variable on the interval $[-0.5, 0.5]$

that is: $X_i$ ~ $U(-0.5, 0.5)$, for $i \in [1, 1500]$

How can I calculate the expected value of the sum $\sum_{i=1}^{1500} X_i$ ?

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    $\begingroup$ expected value is linear $\endgroup$
    – user251257
    Aug 6, 2015 at 10:31

2 Answers 2

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Hint:

In general: $$\mathbb E(X_1+\cdots+X_n)=\mathbb EX_1+\cdots+\mathbb EX_n$$

Provided that all expectations $\mathbb EX_i$ exist.

Also: $$\mathbb EaX=a\mathbb EX$$

Again provided that expectation $\mathbb EX$ exists.

From now on "expectation is linear" should be part of your luggage.

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    $\begingroup$ The Linearity of Expectation is a very useful thing to know. $\endgroup$ Aug 6, 2015 at 10:47
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    $\begingroup$ The theorem on the sum of the mean values ​​apply to every probability distribution of random variables $ X_i $ (also for dependent random variables $ X_i $). $\endgroup$
    – georg
    Aug 6, 2015 at 11:43
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For any $X_i \sim \text{Unif}(-0.5,0.5)$, we have: $$E(X_i) = \frac{-0.5 + 0.5}{2} = 0.$$ The $X_i$ are identically distributed, so this holds for all $i$. By linearity of the expectation operator: \begin{align*} E\left(\sum\limits_{i=1}^{1500} X_i \right) = \sum\limits_{i=1}^{1500} E(X_i) = \sum\limits_{i=1}^{1500} 0 = 1500 \cdot 0 = 0. \end{align*}

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