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In this paper, we consider the Cauchy problem for the generalized KdVequation with rough data and random data. Firstly, we prove that$u(x,t)\longrightarrow u(x,0)$ as $t\longrightarrow0$ for a.e. $x\in\mathbb{R}$ with $u(x,0)\inH^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k},k\geq8).$ Secondly, we prove that$u(x,t)\longrightarrow e^{-t\partial_{x}^{3}}u(x,0)$ as $t\longrightarrow0$ fora.e. $x\in \mathbb{R}$ with $u(x,0)\inH^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k},k\geq8).$ Thirdly, we prove that$\lim\limits_{t\longrightarrow0}\left\|u(x,t)-e^{-t\partial_{x}^{3}}u(x,0)\right\|_{L_{x}^{\infty}}=0$ with$u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k+1},k\geq5)$. Fourthly, byusing Strichartz estimates, probabilistic Strichartz estimates, we establishthe probabilistic well-posedness in $H^{s}(\mathbb{R})\left(s>{\rm max}\left\{\frac{1}{k+1}\left(\frac{1}{2}-\frac{2}{k}\right),\frac{1}{6}-\frac{2}{k}\right\}\right)$ with random data. Our result improvesthe result of Hwang, Kwak (Proc. Amer. Math. Soc. 146(2018), 267-280.).Fifthly, we prove that $\forall \epsilon>0,$ $\forall \omega \in \Omega_{T},$$\lim\limits_{t\longrightarrow0}\left\|u(x,t)-e^{-t\partial_{x}^{3}}u^{\omega}(x,0)\right\|_{L_{x}^{\infty}}=0$with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6),$ where ${\rmP}(\Omega_{T})\geq 1- C_{1}{\rm exp}\left(-\frac{C}{T^{\frac{\epsilon}{48k}}\|u(x,0)\|_{H^{s}}^{2}}\right)$ and$u^{\omega}(x,0)$ is the randomization of $u(x,0)$. Finally, we prove that$\forall \epsilon>0,$ $\forall \omega \in \Omega_{T},\lim\limits_{t\longrightarrow0}\left\|u(x,t)-u^{\omega}(x,0)\right\|_{L_{x}^{\infty}}=0$with ${\rm P}(\Omega_{T})\geq 1- C_{1}{\rm exp}\left(-\frac{C}{T^{\frac{\epsilon}{48k}}\|u(x,0)\|_{H^{s}}^{2}}\right)$ and$u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6)$.

The Cauchy problem for the generalized KdV equation with rough data and random data