Simple stochastic logic gates can compute complex functions using stochastic computing. A stochastic number is encoded by a unary bit stream where each bit is 0 or 1. The value of the number is represented by the percent of 1's in the number, and is interpreted as a probability. Each bit of the stochastic number can be modeled as a Bernoulli random variable, and each stochastic number can be represented by a binomial random variable. The variance of a stochastic number is given by p(1 - p)/N where N represents the number of bits in the sequence, and p represents the mean value of the number. For long word-lengths, a binomial random variable behaves as a Gaussian random variable. The mean and variance of a two-input stochastic logic gate are dependent on the bit-level correlation of the two inputs. This paper derives closed-form expressions for mean and variance of two-input stochastic logic gates with correlated inputs. An approach to synthesize correlated stochastic bit streams with specified correlation from uncorrelated bit streams is also presented. Using the proposed synthesis method, stochastic logic gates are simulated with correlated inputs. The simulated values of means and variances are shown to be the same as the theoretical values; thus, the closed-form expressions are validated.