# Clutter Model in MOT: joint distribution of a random matrix and its column size variable

Suppose $$C_k$$ is a random matrix contains columns of measurement vectors that are random variables: $$C_k=[c_k^1,...,c_k^{m_k^c}]$$ $$m^c_k$$ is the number of columns as well as a random variable. All the measurement vectors are independent.

How we can show that the distribution of $$C_k$$is identical to the joint distribution of $$C_k$$and $$m^c_k$$, i.e. $$p(C_k)=p(C_k,m^c_k)$$?