To overcome the premature convergence caused by the ill-distribution of solutions in the basic Gaussian estimation of distribution algorithm (GEDA), this paper explores a novel GEDA variant with random walk strategies, namely RW-GEDA. In RW-GEDA, the weighted maximum likelihood estimation method is used to estimate the Gaussian distribution. The new candidates are sampled using a shifted mean to enhance exploration performance. When the algorithm stagnates, two random walk strategies, namely, Gaussian random walk and Lévy walk, are activated to enrich the population diversity. Moreover, RW-GEDA is executed in an Eigen coordinate framework to promote the evolution towards the dominant region. The performance of RW-GEDA is evaluated by using the CEC 2014 test suite and compared with other top algorithms from different communities as well as promising GEDA extensions. The statistical results demonstrate the competitive performance of our proposed RW-GEDA in terms of efficiency and accuracy. In addition, RW-GEDA is applied to solve the optimal missile guidance handover problem. To fill the gap in solving this problem, a novel missile guidance advantage model is established, and the optimal missile guidance handover is determined by optimizing the control variables of unmanned combat aerial vehicles. The validity and practicability of the problem model as well as the accuracy and efficiency of RW-GEDA are demonstrated by the experimental results.