Exercise · ZW-EXPECTED-VALUE-010 · ★★★ Advanced
\(X\) has density \(f(x)=\dfrac{1}{\pi(1+x^2)}\) on all of \(\mathbb{R}\). What happens with \(E[X]\)?
Choose one option.
Show answer
- A. It does not exist: \(\int |x|f(x)\,dx\) diverges.
Explanation
The integral of \(|x|f(x)\) is \(\infty\), so the expectation is undefined; symmetry is not enough, since \(\infty-\infty\) is meaningless.
Review the concept: Expected value
The expectation \(E[X]\) of a random variable \(X\) is the average of its possible values, each weighted by its probability.
\[\begin{aligned}E[X]&=\sum_x x\,p(x)\\ E[X]&=\int_{-\infty}^{\infty}x\,f(x)\,dx\end{aligned}\]
More practice problems: expected value
- ZW-EXPECTED-VALUE-001 X takes the values 0, 1 and 2 with probabilities 0.2, 0.5 and 0.3. Compute E[X].
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- ZW-EXPECTED-VALUE-003 X has density f(x)=2x on [0,1] and 0 elsewhere. Compute E[X].
- ZW-EXPECTED-VALUE-004 In a game you win 10 pesos with probability 0.2 and lose 2 with probability 0.8. What is the expect…
- ZW-EXPECTED-VALUE-005 If E[X]=3, what is E[2X+5]?
- ZW-EXPECTED-VALUE-006 With f(x)=2x on [0,1], compute E[X^2].
- ZW-EXPECTED-VALUE-007 Which statement about E[X] is correct?
- ZW-EXPECTED-VALUE-008 X equals -1, 0 or 1 with probabilities tfrac14,tfrac12,tfrac14. Compute E[X^2].