根据定义可知:
\[\begin{eqnarray*}
P(X\leq 0.5) &=& P( X = -1)\\
&=& 0.25
\end{eqnarray*}\]
根据定义可知:
\[\begin{eqnarray*}
P(1.5 < X \leq 2.5) &=& P( X = 2)\\
&=& 0.5
\end{eqnarray*}\]
根据分布函数与分布列的关系可知:
当 \(x<-1\) 时, \(F(x) = P(X\leq x) = 0\) .
当 \(-1 \leq x < 2\) 时, \(F(x) = P(X\leq x) = P(X = -1) = 0.25\) .
当 \(2 \leq x < 3\) 时, \(F(x) = P(X\leq x) = P(X = -1) + P(X=2)= 0.75\) .
当 \(x \geq 3\) 时, \(F(x) = P(X\leq x) = P(X = -1) + P(X=2) + P(X=3)= 1\) .
综上,分布函数为
\[\begin{equation*}
F(x) =
\left\{
\begin{aligned}
& 0, && x < -1,\\
& 0.25, && -1 \leq x < 2,\\
& 0.75, && 2 \leq x < 3,\\
& 1, && x \geq 3.
\end{aligned}
\right.
\end{equation*}\]
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