Exercise · ZW-PROBABILITY-DENSITY-FUNCTION-008 · ★★☆ Intermediate
Which statement about a density \(f\) is true?
Choose one option.
Show answer
- A. It can exceed 1 at a point, but the total area is 1
Explanation
A density is not a probability: for example, \(f(x)=2\) on \([0,\tfrac12]\) is a valid density and equals 2. What must be 1 is the total area.
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-002 If X has density f(x)=2x on [0,1], what is P(X≤ tfrac12)?
- 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-009 X has density f(x)=6x(1-x) on [0,1]. Compute P(tfrac14≤ X≤tfrac34).
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