This paper focuses on developing a particle filter based solution for randomly delayed measurements with an unknown latency probability. A generalized measurement model that includes measurements randomly delayed by an arbitrary but fixed maximum number of time steps along with random packet drops is proposed. Owing to random delays and packet drops in receiving the measurements, the measurement noise sequence becomes correlated. A model for the modified noise is formulated and subsequently its probability density function (pdf) is derived. The recursion equation for the importance weights is developed using pdf of the modified measurement noise in the presence of random delays. Offline and online algorithms for identification of the unknown latency parameter using the maximum likelihood criterion are proposed. Further, this work explores the conditions that ensure the convergence of the proposed particle filter. Finally, three numerical examples, one with a non-stationary growth model and two others with target tracking, are simulated to show the effectiveness and the superiority of the proposed filter over the state-of-the-art.