Correlation between three random variables

Question

Suppose I have three random variables $X,Y,Z$. I know that $X$ and $Y$ are uncorrelated and that $Y$ and $Z$ are uncorrelated, i.e., $\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$ and $\mathbb{E}[YZ]=\mathbb{E}[Y]E[Z]$. By any chance, does this mean that also $X$ and $Z$ are uncorrelated?