What's a “belief” in Bayesian games?

My book defines player $i$s 'belief' about player $j$s type as $p_i(t_j)$.

It then goes on to say that if $t_j$ is presumed to be uniformly distributed on $[0,x]$, then $p_i(t_j) = 1/x$ for all $t_j$.

That makes no sense to me. What do they mean by $1/x$? So if $x = 1$, then $p_i(t_j = 2) = 1$? Even though $2 \notin T_j$??

• $1/x$ is a density not a probability. And if $x=1$ then $0 \le t_j \le 1$ so cannot be $2$ (or is presumed not to be) – Henry Dec 20 '16 at 23:51

$p_i (t_j)$ is the PDF of $i$'s belief over $j$'s types.
Recall that if $t_j$ is uniformly distributed on $[0,x]$, then the probability that $t_j \in [a,b]$, where $0 \leq a \leq b \leq x$ is given by
$$\Pr ( a \leq t_j \leq b) = \int_a^b p_i (t_j) d t_j$$
$$\Pr (t_j = a) = 0$$
for any $a$.