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I'm aware of the conditions of a CDF. to be a cumulative distribution: it is always nonnegative, when x→−∞ x→−∞ it tends to 0, when x→+∞ it tends to 1, and it is right-continuous .

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    $\begingroup$ Note: this is the CDF of the uniform distribution over [0,1], classically denoted by U (0,1). $\endgroup$ – Harry49 Mar 16 '17 at 8:44
  • $\begingroup$ $F(x)=\min(1,\max(0,x))$ works but is not as simple to read or understand as the definition in three pieces $\endgroup$ – Henry Mar 16 '17 at 9:03
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The CDF is proportional to $x$ if it has the form $F(x)=cx$, for some constant $c$. You need that, for $x<0$, $F(x)=0$, and for $x>1$, $F(x)=1$.

What constant $c$ makes it such that $F(0)=0$, $F(1)=1$?

$F(0)=c\cdot0=0$ always.

$F(1)=c\cdot1=1$, so we need $c$ to be equal to 1.

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