# What is the intuitive meaning of $K_1, K_2, K_3$ in regards to the conditional density formula derivation in Brownian motion.

In my text, there is a passage that says:

"Suppose we require the conditional distribution of $X(s)$ given that $X(t) = B$, where $s < t$. The conditional density is: \begin{align*} f_{s\mid t}(x\mid B)&=\frac{f_s(x)f_{t-s}(B-x)}{f_t(B)}\\ &=K_1\exp\left(-\frac{x^2}{2s} - \frac{(B-x^2)}{2(t-s)}\right)\\ &=K_2\exp\left(-x^2\left(\frac{1}{2s}+\frac{1}{2(t-s)}\right)+\frac{Bx}{t-s}\right)\\ &=K_2\exp\left(-\frac{t}{2s(t-s)}\left(x^2-2\frac{sB}{t}x\right)\right)\\ &=K_3\exp\left(-\frac{(x-Bs/t)^2}{2s(t-s)/t}\right), \end{align*}

where $K_1, K_2, K_3$ do not depend on $x$."

And then the text continue on with another subject matter without delving into detail about the meaning of $K_1, K_2, K_3$. What is $K_1, K_2, K_3$. And what is the intuitive logic behind it? I merely want some type of context behind $K_1,K_2,K_3$.

Thanks a lot!

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There is no hidden meaning here, the $K_i$s are normalizing factors necessary to make the integrals of densities equal 1 and whose precise values are irrelevant. This explains why the author did not bother to compute them. – Did Dec 5 '12 at 20:07