Exercise · ZW-CUMULATIVE-DISTRIBUTION-FUNCTION-005 · ★★☆ Intermediate
Which of these functions , with the stated values outside the interval, can NOT be a cumulative distribution function ?
Choose one option.
A \(1-e^{-x}\) for \(x\ge 0\), and 0 if \(x<0\) B \(1-x^2\) for \(0\le x\le 1\), with 1 if \(x<0\) and 0 if \(x>1\) C \(x^2\) for \(0\le x\le 1\), with 0 if \(x<0\) and 1 if \(x>1\) D A staircase with jumps 0.4 at 0 and 0.6 at 1 Show answer B. \(1-x^2\) for \(0\le x\le 1\), with 1 if \(x<0\) and 0 if \(x>1\)← Back to the question
Explanation A distribution function never decreases, equals 0 at \(-\infty\) and 1 at \(+\infty\). \(1-x^2\) falls from 1 to 0, so it fails; the others satisfy all three properties.
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 \(x\).
\[F_X(x)=P(X\le x)\]
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