On the Distribution of the Number of Goldbach Partitions of a Randomly Chosen Positive Even Integer

Let $\mathcal{P}=\{p_1,p_2,...\}$ be the set of all odd primes arranged in increasing order. A Goldbach partition of the even integer $2k>4$ is a way of writing it as a sum of two primes from $\mathcal{P}$ without regard to order. Let $Q(2k)$ be the number of all Goldbach partitions of the number $2k$. Assume that $2k$ is selected uniformly at random from the interval $(4,2n], n>2$, and let $Y_n=Q(2k)$ with probability $1/(n-2)$. We prove that the random variable $\frac{Y_n}{n/\left(\frac{1}{2}\log{n}\right)^2}$ converges weakly, as $n\to\infty$, to a uniformly distributed random variable in the interval $(0,1)$. The method of proof uses size-biasing and the Laplace transform continuity theorem.