Exercise · ZW-PROBABILITY-DENSITY-FUNCTION-002 · ★☆☆ Basic
If \(X\) has density \(f(x)=2x\) on \([0,1]\), what is \(P(X\le \tfrac12)\)?
Choose one option.
Show answer
- A. \(\tfrac14\)
Explanation
Integrate the density: \(\int_0^{1/2}2x\,dx=\left(\tfrac12\right)^2=\tfrac14\). Common mistake: thinking half the interval has probability \(\tfrac12\).
Review the concept: Probability density function
The density function \(f\) of a continuous random variable \(X\) is a nonnegative function such that the area under its graph over an interval is the probability \(P\) that \(X\) falls in it.
\[P(a\le X\le b)=\int_a^b f(x)\,dx\]
More practice problems: probability density function
- ZW-PROBABILITY-DENSITY-FUNCTION-001 A continuous variable has density f(x)=kx for 0≤ x≤ 2 and 0 elsewhere. What is k?
- ZW-PROBABILITY-DENSITY-FUNCTION-003 X has constant density f(x)=tfrac14 on [0,4]. What is P(1≤ X≤ 3)?
- ZW-PROBABILITY-DENSITY-FUNCTION-004 If X is continuous with density f, what is P(X=2)?
- ZW-PROBABILITY-DENSITY-FUNCTION-005 X has density f(x)=3x^2 on [0,1]. Compute P(X>tfrac12).
- ZW-PROBABILITY-DENSITY-FUNCTION-006 X has density f(x)=e^-x for x≥ 0. What is P(X>2)?
- ZW-PROBABILITY-DENSITY-FUNCTION-007 Which of these functions, equal to 0 outside its interval, is a density?
- ZW-PROBABILITY-DENSITY-FUNCTION-008 Which statement about a density f is true?
- ZW-PROBABILITY-DENSITY-FUNCTION-009 X has density f(x)=6x(1-x) on [0,1]. Compute P(tfrac14≤ X≤tfrac34).
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