Track / Stage 5 / 133 of 400 #733 of 1964

Problem 733

AIME late

Algebra Difficulty 5.3 Find the answer

1. Let the set $A=\left\{x \left\lvert\, \frac{x+4}{x-3} \leqslant 0\right., x \in \mathbf{Z}\right\}$ , and from the set $A$ a random element $x$ is drawn, denoted by $\xi=x^{2}$ . Then the mathematical expectation of the random variable $\xi$ is $\mathrm{E} \xi=$ $\qquad$

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

$-, 1.5$
From the conditions, we know
$A=\{-4,-3,-2,-1,0,1,2\},$

The values of the random variable $\xi$ are $0,1,4,9,16$ .
It is easy to see that the probability distribution of $\xi$ is shown in Table 1. Therefore, E $\xi$
$\begin{array}{l} =0 \times \frac{1}{7}+1 \times \frac{2}{7}+4 \times \frac{2}{7}+9 \times \frac{1}{7}+16 \times \frac{1}{7} \\ =5 \end{array}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.