Exercise · ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-010 · ★★★ Advanced
\(F(x)=1-e^{-2x}\) for \(x\ge 0\). For which value \(m\) does \(P(X\le m)=\tfrac12\) hold?
Choose one option.
Show answer
- A. \(\tfrac{\ln 2}{2}\)
Explanation
Solve \(1-e^{-2m}=\tfrac12\), i.e. \(e^{-2m}=\tfrac12\), so \(m=\tfrac{\ln 2}{2}\approx 0.347\).
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-007 If F(x)=x^3 for 0≤ x≤ 1, what is the density at x=tfrac12?
- 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…