Traditional reliability analysis uses probability distributions to calculate reliability. In many engineering applications, some nondeterministic variables are known within intervals. When both random variables and interval variables are present, a single probability measure, namely, the probability of failure or reliability, is not available in general; but its lower and upper bounds exist. The mixture of distributions and intervals makes reliability analysis more difficult. Our goal is to investigate computational tools to quantify the effects of random and interval inputs on reliability associated with performance characteristics. The proposed reliability analysis framework consists of two components — direct reliability analysis and inverse reliability analysis. The algorithms are based on the First Order Reliability Method and many existing reliability analysis methods. The efficient and robust improved HL-RF method is further developed to accommodate interval variables. To deal with interval variables for black-box functions, nonlinear optimization is used to identify the extreme values of a performance characteristic. The direct reliability analysis provides bounds of a probability of failure; the inverse reliability analysis computes the bounds of the percentile value of a performance characteristic given reliability. One engineering example is provided.