Probability Model of Directed Movements Based on Total Probability Theorem

When the locations of an agent at two times,and its maximum velocity are known,the agent’s location between both those time instances is uncertain.We present a practical method,the total probability theorem,to approximate that uncertainty.First,the minimum(average) velocity from starting point to destination can be computed,and then many discrete speed values between the minimum and maximum velocity can be chosen randomly.The random speed variable V follows the Maxwell-Boltzmann distribution that describes particle speeds,and thus the probability density function of V,p(V),becomes applicable.Second,for a discrete speed value v,we calculate the agent’s reachable range(x,y) at any time t in time geography.The range follows a uniform distribution,and so at t we may obtain p(x,y | v,t),which is the conditional probability of(x,y) given the value of the random variable V,V=v.Finally,according to the total probability theorem,the probability distribution of the agent at time t,p(x,y|t),is obtained by the equation ∑p(V=v)·p(x,y | v,t) where the parameter V takes all values.When increasing the maximum velocity,experiments show that the total probability’ variance has a good convergence and steadiness,an improvement over the existing method’ divergence.