Exercise · ZW-VARIANCE-010 · ★★★ Advanced
\(X\) has density \(f(x)=\tfrac14\) on \([0,4]\). Compute \(\operatorname{Var}(X)\).
Choose one option.
Show answer
- B. \(\dfrac{4}{3}\)
Explanation
\(E[X]=2\) and \(E[X^2]=\tfrac{16}{3}\), so \(\operatorname{Var}(X)=\tfrac{16}{3}-4=\tfrac43\). The value \(\tfrac{16}{3}\) forgets to subtract \((E[X])^2\).
Review the concept: Variance
The variance \(\operatorname{Var}(X)\) of a random variable \(X\) measures how spread out its values are around its expectation \(E\): it is the expected value of the squared deviation.
\[\operatorname{Var}(X)=E\big[(X-E[X])^2\big]\]
More practice problems: variance
- ZW-VARIANCE-001 X takes the values 0, 1 and 2 with probabilities 0.2, 0.5 and 0.3 (so E[X]=1.1). Compute Var(X).
- ZW-VARIANCE-002 Compute the variance of the outcome of a fair die.
- ZW-VARIANCE-003 X has density f(x)=2x on [0,1]. Compute Var(X).
- ZW-VARIANCE-004 If Var(X)=4, what is Var(3X+1)?
- ZW-VARIANCE-005 If E[X]=2 and E[X^2]=7, what is Var(X)?
- ZW-VARIANCE-006 X equals 1 with probability 0.3 and 0 with probability 0.7. Compute Var(X).
- ZW-VARIANCE-007 Which statement about the variance is correct?
- ZW-VARIANCE-008 X equals -2 or 2 with probability tfrac12 each. Compute Var(X).