Exercise · ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-007 · ★★☆ Intermediate
If \(F(x)=x^3\) for \(0\le x\le 1\), what is the density at \(x=\tfrac12\)?
Choose one option.
Show answer
- D. \(\tfrac34\)
Explanation
The density is the derivative: \(f(x)=F'(x)=3x^2\), and \(f(\tfrac12)=\tfrac34\). The option \(\tfrac18\) is \(F(\tfrac12)\).
Review the concept: Cumulative distribution function
The cumulative distribution function \(F_X\) of a random variable \(X\) gives, for each real number \(x\), the probability \(P\) that \(X\) takes a value less than or equal to \(x\).
\[F_X(x)=P(X\le x)\]
More practice problems: cumulative distribution function
- ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-001 X takes the values 0, 1 and 2 with probabilities 0.2, 0.5 and 0.3. What is F(1)?
- ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-002 If F(2)=0.8, what is P(X>2)?
- ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-003 X has density f(x)=2x on [0,1]. What is F(x) for 0≤ x≤ 1?
- ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-004 It is known that F(3)=0.9 and F(1)=0.35. What is P(1<X≤ 3)?
- ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-005 Which of these functions, with the stated values outside the interval, can NOT be a cumulative dist…
- ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-006 If X has density f(x)=e^-x for x≥ 0, what is P(1<X≤ 2)?
- ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-008 For a discrete variable, F(0)=0.2, F(1)=0.5 and F(2)=1, and it only takes the values 0, 1 and 2. Wh…
- ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-009 Let F(x)=(x^2+x)/(6) for 0≤ x≤ 2 (0 before and 1 after). What is P(X>1)?
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